(As per the syllabus of the Council for the Indian School Certificate Examinations)

1. Introduction

We see many objects around us that have length, breadth, and height. Such objects are called three-dimensional (3D) shapes or solid shapes.
Examples: book, dice, ball, box, ice-cream cone, etc.

Unlike 2D shapes (square, rectangle, circle) which lie on a flat surface, 3D shapes occupy space.

2. Faces, Edges, and Vertices

Every 3D shape has three basic parts:

Face: Flat or curved surface of a solid
Edge: Line where two faces meet
Vertex (plural: Vertices): Point where edges meet

Example:

A cube has:

Faces = 6
Edges = 12
Vertices = 8

3. Common Three-Dimensional Shapes

(a) Cube

All faces are square
All edges are equal
Examples: dice, ice cube

Properties of a Cube

Faces = 6
Edges = 12
Vertices = 8

(b) Cuboid

Faces are rectangular
Length, breadth, and height may be different
Examples: book, brick, matchbox

Properties of a Cuboid

Faces = 6
Edges = 12
Vertices = 8

(c) Sphere

Completely round
Has only one curved face
No edges and no vertices
Example: football, globe

Properties

Faces = 1 (curved)
Edges = 0
Vertices = 0

(d) Cylinder

Has two flat circular faces and one curved surface
Examples: gas cylinder, tin can

Properties

Faces = 3 (2 flat + 1 curved)
Edges = 2
Vertices = 0

(e) Cone

Has one flat circular base and one curved surface
Has one vertex (top point)
Examples: ice-cream cone, funnel

Properties

Faces = 2
Edges = 1
Vertices = 1

(f) Hemisphere

Half of a sphere
Has one flat face and one curved face
Example: bowl

4. Polyhedrons

A polyhedron is a 3D shape:

Made only of flat faces
Faces are polygons

Examples:

Cube
Cuboid
Prism
Pyramid

Sphere, cone, and cylinder are NOT polyhedrons because they have curved surfaces.

5. Nets of Three-Dimensional Shapes

A net is a flat pattern that can be folded to form a 3D shape.

Examples:

A cube net has 6 squares
A cuboid net has 6 rectangles
A cone net has 1 circle + 1 sector
A cylinder net has 2 circles + 1 rectangle

Nets help us understand:

Surface area
Shape structure

6. Euler’s Formula

For many polyhedrons:

[
\text{Faces (F)} + \text{Vertices (V)} = \text{Edges (E)} + 2
]

Example (Cube):

F = 6
V = 8
E = 12

[
6 + 8 = 12 + 2 \quad \checkmark
]

7. Difference Between 2D and 3D Shapes

2D Shapes	3D Shapes
Flat shapes	Solid shapes
Have length & breadth	Have length, breadth & height
No volume	Have volume
Example: square	Example: cube

8. Real-Life Examples of 3D Shapes

Object	Shape
Dice	Cube
Book	Cuboid
Ball	Sphere
Tin can	Cylinder
Ice-cream cone	Cone

9. Important Points to Remember

3D shapes occupy space
Faces can be flat or curved
Edges are straight lines
Vertices are corner points
Nets help in visualizing solids

10. Exam-Oriented Tips

Learn faces, edges, vertices of each shape
Practice nets
Remember Euler’s formula
Use real-life examples in answers
Draw neat diagrams in exams

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✅ Short notes for revision
✅ One-page summary
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Below are detailed Class 7 ICSE Mathematics notes on “Three Dimensional Shapes” written in a long, easy-to-understand format (approx. 5000-word style explanation) suitable for school study and revision.

Class 7 ICSE Mathematics

Three Dimensional Shapes – Detailed Notes

1. Introduction to Three Dimensional Shapes

In mathematics, shapes can be classified into two major types:

Two Dimensional Shapes (2D Shapes)
Three Dimensional Shapes (3D Shapes)

Two Dimensional Shapes

Two dimensional shapes have only length and breadth. They are flat shapes that lie on a plane surface.

Examples:

Square
Rectangle
Triangle
Circle

These shapes do not have thickness or height.

Three Dimensional Shapes

Three dimensional shapes have length, breadth, and height (or depth). Because they have thickness, they occupy space.

Such shapes are also called solid shapes or solids.

Examples of 3D shapes around us include:

A dice which has the shape of a cube.
A brick which has the shape of a cuboid.
A football which has the shape of a sphere.
A water bottle which has the shape of a cylinder.
An ice cream cone which has the shape of a cone.

Three dimensional shapes are important because they represent the objects we see in real life.

2. Understanding Space and Solid Figures

When we talk about three dimensional shapes, we are dealing with space. Space means the region in which objects exist.

A solid figure occupies space and has volume.

For example:

A box occupies space.
A ball occupies space.
A bottle occupies space.

The amount of space inside a solid object is called volume.

The outer covering of a solid object is called its surface.

3. Parts of Three Dimensional Shapes

Three dimensional shapes are made up of different parts. These parts are important to understand the structure of solids.

The main parts are:

Faces
Edges
Vertices

3.1 Face

A face is a flat surface of a three dimensional shape.

Examples:

A cube has six square faces.
A cuboid has six rectangular faces.

Faces can be:

Flat faces
Curved faces

For example:

Cube → flat faces
Sphere → curved surface

3.2 Edge

An edge is the line segment where two faces meet.

Example:

A cube has 12 edges.
A cuboid also has 12 edges.

Edges help in forming the structure of the solid.

3.3 Vertex (Plural: Vertices)

A vertex is the point where three or more edges meet.

Example:

A cube has 8 vertices.
A cuboid also has 8 vertices.

Vertices are the corner points of solid shapes.

4. Polyhedron

A polyhedron is a three dimensional solid made up of flat polygonal faces.

Examples:

Cube
Cuboid
Pyramid
Prism

The word polyhedron means many faces.

Characteristics of a polyhedron:

Faces are polygons.
Edges join the faces.
Vertices are formed by the meeting of edges.

Shapes like sphere, cylinder, and cone are not polyhedrons because they have curved surfaces.

5. Euler’s Formula

A famous rule that applies to many polyhedrons is Euler’s Formula.

The formula is:

F + V − E = 2

Where:

F = Number of faces
V = Number of vertices
E = Number of edges

Example: Cube

Faces = 6
Vertices = 8
Edges = 12

Applying the formula:

6 + 8 − 12 = 2

Hence the formula is correct.

Euler’s formula helps in checking whether the values of faces, edges and vertices are correct.

6. Important Three Dimensional Shapes

The most important 3D shapes studied in Class 7 are:

Cube
Cuboid
Sphere
Cylinder
Cone
Prism
Pyramid

Let us study them one by one.

7. Cube

A cube is a three dimensional shape in which all faces are squares and all edges are equal.

Properties of Cube

Number of faces = 6
Number of edges = 12
Number of vertices = 8
All faces are squares
All edges are equal

Real Life Examples

Dice
Ice cube
Rubik’s cube
Sugar cube

Surface Area of Cube

The surface area is the total area covered by all faces of the cube.

Formula:

Surface Area = 6a²

Where
a = side of the cube

Example:

Side = 5 cm

Surface area = 6 × 5²
= 6 × 25
= 150 cm²

Volume of Cube

Volume means the space occupied by the cube.

Formula:

Volume = a³

Example:

Side = 5 cm

Volume = 5³
= 125 cm³

8. Cuboid

A cuboid is a three dimensional shape with rectangular faces.

A cuboid looks like a box.

Properties of Cuboid

Faces = 6
Edges = 12
Vertices = 8
Opposite faces are equal

Examples

Brick
Book
Pencil box
Matchbox

Surface Area of Cuboid

Formula:

Total Surface Area

= 2(lb + bh + lh)

Where

l = length
b = breadth
h = height

Volume of Cuboid

Volume = l × b × h

Example:

l = 10 cm
b = 5 cm
h = 4 cm

Volume = 10 × 5 × 4
= 200 cm³

9. Sphere

A sphere is a perfectly round three dimensional shape.

Every point on the surface of a sphere is at the same distance from the centre.

Examples

Football
Globe
Marble
Ball

Properties

One curved surface
No edges
No vertices

Surface Area of Sphere

Formula:

Surface Area = 4πr²

Where
r = radius

Volume of Sphere

Formula:

Volume = 4/3 πr³

10. Cylinder

A cylinder has two circular bases connected by a curved surface.

Examples

Gas cylinder
Water bottle
Tin can
Pipe

Properties

Two circular faces
One curved surface
No vertices

Curved Surface Area

Formula:

CSA = 2πrh

Where
r = radius
h = height

Total Surface Area

Formula:

TSA = 2πr(r + h)

Volume of Cylinder

Formula:

Volume = πr²h

11. Cone

A cone is a three dimensional shape with one circular base and one curved surface that meets at a point.

Examples

Ice cream cone
Birthday cap
Traffic cone

Properties

One circular base
One curved surface
One vertex

Slant Height

The slant height is the distance from the top vertex to the edge of the circular base.

Curved Surface Area

Formula:

CSA = πrl

Where
l = slant height

Total Surface Area

Formula:

TSA = πr(r + l)

Volume of Cone

Formula:

Volume = 1/3 πr²h

12. Prism

A prism is a three dimensional shape with two identical bases and rectangular faces.

Examples:

Triangular prism
Rectangular prism

Properties

Two parallel bases
Lateral faces are rectangles
Cross section is constant

13. Pyramid

A pyramid is a solid shape with a polygon base and triangular faces meeting at a single point.

Examples:

Egyptian pyramids
Square pyramid

14. Nets of Three Dimensional Shapes

A net is a pattern formed by unfolding a 3D shape into a 2D shape.

When folded back, it forms the solid.

Examples:

Cube Net → 6 squares
Cuboid Net → 6 rectangles
Cylinder Net → 2 circles + 1 rectangle

Nets help us understand how surfaces form a solid.

15. Cross Sections

A cross section is the shape obtained when a solid is cut by a plane.

Examples:

Cutting a cylinder horizontally → circle
Cutting a cone vertically → triangle
Cutting a cube → square or rectangle

16. Real Life Applications

Three dimensional shapes are used everywhere in daily life.

Examples:

Architecture
Buildings are designed using cuboids and prisms.

Engineering
Machines contain cylindrical parts.

Packaging
Boxes are cuboids.

Sports
Balls are spheres.

Food
Ice cream cones are cones.

17. Comparison of 3D Shapes

Shape	Faces	Edges	Vertices
Cube	6	12	8
Cuboid	6	12	8
Cylinder	3 surfaces	2 edges	0
Cone	2 surfaces	1 edge	1
Sphere	1 surface	0	0

18. Example Problems

Question 1

Find the volume of a cube with side 6 cm.

Solution

Volume = a³
= 6³
= 216 cm³

Question 2

Find the volume of a cuboid with

l = 12 cm
b = 5 cm
h = 4 cm

Volume = l × b × h
= 12 × 5 × 4
= 240 cm³

Question 3

Find the curved surface area of a cylinder.

r = 7 cm
h = 10 cm

CSA = 2πrh

= 2 × 22/7 × 7 × 10
= 440 cm²

19. Importance of Three Dimensional Geometry

Three dimensional geometry is important because:

It helps understand real world objects.
It is useful in architecture and engineering.
It is used in science and technology.
It improves spatial thinking.

Students learn how objects occupy space and how their surfaces and volumes are calculated.

20. Conclusion

Three dimensional shapes are an important part of mathematics. They help us understand the objects that exist in the real world. By studying solids like cube, cuboid, cylinder, cone and sphere, students learn about their faces, edges, vertices, surface areas and volumes.

Understanding these shapes also helps in solving practical problems related to packaging, construction, engineering and design.

Thus, the study of three dimensional shapes builds a strong foundation for higher mathematics and real life applications.

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Class 7 ICSE Mathematics

Three Dimensional Shapes – Extended Notes

21. Difference Between 2D Shapes and 3D Shapes

Understanding the difference between two-dimensional and three-dimensional shapes is very important.

Feature	2D Shapes	3D Shapes
Dimensions	Length and Breadth	Length, Breadth and Height
Thickness	No thickness	Has thickness
Occupies Space	No	Yes
Volume	No volume	Has volume
Examples	Square, Triangle, Circle	Cube, Sphere, Cylinder

Example

A square drawn on paper is a 2D shape, but a dice is a 3D shape.

22. Solid Shapes in Daily Life

Three dimensional shapes are found everywhere around us. Understanding these shapes helps us recognize objects in daily life.

Examples:

Object	Shape
Dice	Cube
Brick	Cuboid
Ball	Sphere
Water tank	Cylinder
Ice cream	Cone
Pyramid monument	Pyramid

Recognizing these shapes helps students relate mathematics to the real world.

23. Curved Surface and Flat Surface

Surfaces of solids can be of two types.

Flat Surface

A flat surface is a plane surface.

Examples:

Cube
Cuboid
Prism

Curved Surface

A curved surface is not flat.

Examples:

Sphere
Cylinder
Cone

Some shapes have both flat and curved surfaces.

Example: Cylinder → flat circular faces + curved surface.

24. Lateral Surface Area

The lateral surface area of a solid is the area of all the sides excluding the base and top.

Example:

Cylinder
Lateral surface = curved surface only

Formula

Lateral Surface Area = 2πrh

Cone
Lateral surface = curved surface

Formula

Lateral Surface Area = πrl

25. Surface Area

The surface area of a solid is the total area of all its outer surfaces.

Surface area helps us calculate:

Paint needed for a wall
Wrapping paper required
Material used in making containers

Example:

Surface area of cube = 6a²

26. Volume

Volume is the amount of space occupied by a solid object.

Unit of volume:

cubic centimetre (cm³)
cubic metre (m³)

Example:

If a cube has side 3 cm

Volume = 3 × 3 × 3
= 27 cm³

Volume helps in measuring capacity of containers.

27. Capacity

Capacity refers to the amount of liquid a container can hold.

Example:

Water bottle
Tank
Glass

Units of capacity:

millilitre (mL)
litre (L)

Relation:

1 litre = 1000 cm³

28. Cross Sections of Solids

A cross section is the shape obtained when a solid is cut by a plane.

Examples:

Cube cut parallel to base → square

Cylinder cut horizontally → circle

Cone cut vertically → triangle

Cross sections are used in engineering and architecture.

29. Prism in Detail

A prism is a solid with:

two identical bases
rectangular side faces

Example:
Triangular prism.

Properties of Prism

Bases are parallel
Side faces are rectangles
Cross section remains the same

Examples in real life:

Glass prism
Toblerone chocolate box

30. Pyramid in Detail

A pyramid has:

one base
triangular side faces
one vertex at the top

Types of pyramids:

Triangular pyramid
Square pyramid
Pentagonal pyramid

Real Life Example

The famous pyramids of Egypt.

31. Regular and Irregular Solids

Regular Solid

A regular solid has all faces equal.

Example: Cube

Irregular Solid

Faces are not equal.

Example: Cuboid

32. Hollow Solids

Some solids are hollow instead of solid.

Example:

Pipe
Tube
Water tank

These shapes have outer surface area and inner surface area.

33. Relationship Between Radius, Diameter and Height

Radius

Distance from center to edge of circle.

Diameter

Diameter = 2 × radius

Height

Vertical distance between top and base.

These measurements are used in formulas for cylinder, cone, and sphere.

34. Slant Height of Cone

The slant height of a cone is the distance from the vertex to the edge of the base.

Formula:

l² = r² + h²

Where

l = slant height
r = radius
h = height

This formula comes from the Pythagoras theorem.

35. Units Used in Three Dimensional Geometry

Measurements are expressed using standard units.

Length

mm
cm
m
km

Area

cm²
m²

Volume

cm³
m³

Capacity

mL
L

36. Conversion of Units

Important conversions:

1 m = 100 cm
1 cm = 10 mm

Area conversions:

1 m² = 10000 cm²

Volume conversions:

1 m³ = 1000000 cm³

Capacity conversion:

1 litre = 1000 mL

37. Real Life Applications of 3D Shapes

Three dimensional shapes are used in many fields.

Architecture

Buildings are constructed using cuboids and prisms.

Engineering

Machines contain cylindrical and spherical parts.

Packaging Industry

Boxes and containers are cuboids.

Sports

Balls used in sports are spheres.

Household Items

Water tanks are cylinders.

38. Visualization of Solids

Visualization means imagining objects in space.

Students should practice identifying shapes from different views:

top view
side view
front view

This helps improve spatial understanding.

39. Symmetry in Three Dimensional Shapes

Many solids have symmetry.

Example:

Cube has many lines of symmetry.

Sphere has infinite symmetry because it looks the same from every direction.

Symmetry helps in design and engineering.

40. Common Mistakes Students Make

Students often make mistakes in:

Using wrong formulas
Forgetting units
Confusing surface area with volume
Using diameter instead of radius
Incorrect calculations with π

To avoid mistakes:

Write formulas clearly
Check units
Recheck calculations

41. Exam Tips for This Chapter

For ICSE exams:

Learn formulas properly.
Practice numerical questions.
Draw diagrams neatly.
Write units in answers.
Show full steps in calculations.

42. Practice Questions

Question 1

Find the surface area of a cube of side 7 cm.

Question 2

Find the volume of a cuboid of dimensions:

12 cm × 8 cm × 5 cm.

Question 3

Find the curved surface area of a cylinder:

r = 7 cm
h = 10 cm.

Question 4

Find the volume of a cone:

r = 7 cm
h = 12 cm.

Question 5

Find the surface area of a sphere:

r = 14 cm.

43. Higher Thinking Questions

Why does a sphere have no edges?
Why are cubes used in dice?
Why are water tanks cylindrical?

These questions develop logical thinking.

44. Summary of Important Formulas

Cube

Surface Area = 6a²
Volume = a³

Cuboid

Surface Area = 2(lb + bh + lh)
Volume = lbh

Cylinder

CSA = 2πrh
Volume = πr²h

Cone

CSA = πrl
Volume = 1/3 πr²h

Sphere

Surface Area = 4πr²
Volume = 4/3 πr³

45. Conclusion

Three dimensional shapes are an essential part of geometry. They help us understand how objects exist in space and how their surfaces and volumes are measured. Through this chapter students learn about the structure, properties and formulas related to various solids such as cubes, cuboids, cylinders, cones and spheres.

The knowledge of three dimensional shapes is not only useful for solving mathematical problems but also plays an important role in real life applications such as construction, engineering, packaging and design.

If you want, I can also give you:

20 ICSE exam-type questions with solutions
Extra long 7000–8000 word notes
Important diagrams of 3D shapes for exams
Full ICSE exercise solutions of this chapter.

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Class 7 ICSE Mathematics

Three Dimensional Shapes – Additional Detailed Notes

46. Classification of Three Dimensional Shapes

Three dimensional shapes can be classified into two main groups:

1. Polyhedrons

2. Non-Polyhedrons

Polyhedrons

A polyhedron is a solid shape made entirely of flat polygonal faces.

Examples:

Cube
Cuboid
Prism
Pyramid

Characteristics of polyhedrons:

Faces are flat polygons
Edges are straight lines
Vertices are corner points

Non-Polyhedrons

These shapes have curved surfaces.

Examples:

Sphere
Cylinder
Cone

They do not satisfy the conditions of polyhedrons.

47. Regular Polyhedrons

A regular polyhedron is a solid where:

All faces are equal polygons
All edges are equal
All angles are equal

Example:

Cube

The cube is one of the most common regular polyhedrons.

48. Irregular Polyhedrons

If the faces or edges are not equal, the solid is called an irregular polyhedron.

Example:

Cuboid

In a cuboid, the faces are rectangles and may not all be equal.

49. Open and Closed Solids

Closed Solid

A solid whose surface completely encloses space.

Examples:

Ball
Box
Water bottle

Open Solid

A solid that does not fully enclose space.

Example:

A hollow pipe with open ends.

50. Transparent and Opaque Solids

Transparent Solids

Objects through which light can pass.

Example:

Glass prism

Opaque Solids

Objects that do not allow light to pass.

Example:

Wooden box

51. Solids Formed by Rotation

Some 3D shapes can be formed when 2D shapes rotate around a line.

Examples:

Rectangle rotating around one side → Cylinder

Right triangle rotating around one side → Cone

Semicircle rotating around diameter → Sphere

This concept is called solid of revolution.

52. Views of Three Dimensional Shapes

When we look at an object, we see it from different directions.

Types of Views

Top View
Front View
Side View

Example:

A cube from the top looks like a square.

This concept is useful in engineering drawings.

53. Projection of Solids

Projection means representing a 3D object on a 2D plane.

Architects and engineers use projection to draw building plans.

Example:

Floor plans
Machine designs

54. Development of Solids

Development means opening a solid into a flat shape.

This flat shape is called the net.

Example:

A cube when opened forms six connected squares.

This concept is used in:

box making
packaging
carton design

55. Edge Length and Diagonal

Edge

Edge is the line segment joining two vertices.

Example:

Cube has 12 edges.

Diagonal

A diagonal is a line joining two opposite vertices.

Example:

Cube has body diagonals.

56. Space Diagonal of Cuboid

The diagonal inside a cuboid can be calculated using the formula:

d = √(l² + b² + h²)

Where

d = diagonal
l = length
b = breadth
h = height

This formula comes from the Pythagoras theorem.

57. Packing of Solids

Packing means arranging solids in space efficiently.

Example:

Cubes pack perfectly without leaving gaps.

But spheres leave empty spaces.

This concept is used in:

storage
transportation
packaging industry

58. Stacking of Cubes

Sometimes cubes are stacked together.

Example problem:

If 27 small cubes are stacked to form a big cube.

Side of big cube = 3 cubes

Volume relationship:

Total cubes = 3 × 3 × 3 = 27

59. Painting Cubes Problems

These problems are common in exams.

Example:

A cube is painted on all sides and cut into smaller cubes.

Questions asked:

How many cubes have 3 faces painted?
How many cubes have 2 faces painted?
How many cubes have 1 face painted?

Example:

If cube is cut into 3 × 3 × 3 = 27 cubes

Corner cubes = 8

These have 3 faces painted.

60. Surface Area in Daily Life

Surface area is important in many real life situations.

Examples:

Painting walls
Surface area tells how much paint is required.

Wrapping gifts
Surface area tells how much wrapping paper is needed.

Making tents
Surface area helps calculate fabric required.

61. Volume in Daily Life

Volume is used in many real life calculations.

Examples:

Water tank capacity
Fuel tanks
Storage containers

Example:

A cylindrical water tank stores water according to its volume.

62. Capacity of Containers

Capacity refers to maximum liquid a container can hold.

Examples:

Bottle
Bucket
Tank

Relation:

1 cubic centimetre = 1 millilitre

63. Importance of π (Pi)

Many formulas use π (pi).

Value of π:

π ≈ 22/7

π ≈ 3.14

It is used in calculations involving circles and curved surfaces.

64. Steps to Solve Numerical Problems

When solving questions:

Write the given values.
Write the formula.
Substitute the values.
Solve step by step.
Write correct units.

Example:

Find volume of cube with side 4 cm.

Volume = a³
= 4³
= 64 cm³

65. Sample ICSE Exam Problems

Problem 1

Find the volume of a cuboid:

l = 10 cm
b = 6 cm
h = 4 cm

Solution:

Volume = l × b × h
= 10 × 6 × 4
= 240 cm³

Problem 2

Find the surface area of a cube of side 8 cm.

Solution:

Surface area = 6a²
= 6 × 8²
= 6 × 64
= 384 cm²

Problem 3

Find the curved surface area of a cylinder.

r = 7 cm
h = 12 cm

CSA = 2πrh

= 2 × 22/7 × 7 × 12
= 528 cm²

66. Higher Level Thinking

Students should think about questions like:

Why are most storage tanks cylindrical?
Why are balls spherical?
Why are bricks cuboid?

These questions help connect math with real life.

67. Activities for Students

Teachers may ask students to perform activities.

Example activities:

Make a cube using paper.
Create nets of 3D shapes.
Identify shapes in classroom objects.

These activities help students visualize shapes better.

68. Use of Three Dimensional Geometry in Science

Three dimensional shapes are used in science fields.

Examples:

Chemistry
Molecules have 3D structures.

Physics
Objects move in three dimensional space.

Astronomy
Planets are spherical.

69. Use in Technology

3D shapes are used in modern technology.

Examples:

3D printing
Computer graphics
Animation and gaming

Three dimensional modelling helps create realistic designs.

70. Final Summary of the Chapter

Three dimensional shapes are solid figures that occupy space. They have length, breadth and height. These shapes include cube, cuboid, cylinder, cone, sphere, prism and pyramid. Each shape has unique properties such as faces, edges and vertices.

Students learn how to calculate surface area and volume of these shapes using formulas. These calculations are useful in real life situations like construction, packaging and manufacturing.

Understanding three dimensional geometry helps students develop spatial thinking and problem solving skills.

If you want, I can also make:

Class 7 ICSE Three Dimensional Shapes – 50 Exam Questions with Answers
Very short 2-page revision notes for exams
Important diagrams of cube, cuboid, cylinder, cone, sphere
Full ICSE exercise solutions of this chapter.

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Class 7 ICSE Mathematics

Three Dimensional Shapes – Further Extended Notes

71. Historical Background of Three Dimensional Geometry

The study of shapes and solids is very old. Ancient mathematicians studied three dimensional figures to understand the world around them.

One of the most famous mathematicians who worked on geometry was . He wrote an important book called , which explained many principles of geometry.

Another important mathematician was , who introduced Euler’s Formula for polyhedrons.

These mathematicians helped develop the geometry that students study today.

72. Importance of Three Dimensional Shapes in Architecture

Architecture is the art of designing buildings. Architects use three dimensional shapes to plan and construct structures.

Examples:

Buildings often use cuboids as the basic structure.
Domes of buildings may have spherical shapes.
Towers may have cylindrical shapes.

Famous structures around the world use three dimensional geometry.

For example, the is a large pyramid-shaped structure built thousands of years ago.

This shows how important geometric shapes are in construction.

73. Three Dimensional Shapes in Nature

Nature also contains many examples of three dimensional shapes.

Examples:

Fruits like oranges resemble spheres.
Honeycombs have prism-like structures.
Mountains may resemble pyramids.

Even the planets in our solar system are nearly spherical in shape, such as .

74. Three Dimensional Shapes in Art and Design

Artists and designers use three dimensional shapes to create sculptures and models.

Examples:

Clay models
Wooden carvings
Paper crafts

Many sculptures are based on shapes like cubes, cylinders, and cones.

Three dimensional art helps create realistic and visually appealing designs.

75. Three Dimensional Shapes in Engineering

Engineering involves designing machines and structures.

Engineers often use basic geometric shapes to design machine parts.

Examples:

Pipes are cylindrical.
Balls in ball bearings are spherical.
Storage tanks are cylindrical.

These shapes are used because they are strong and efficient.

76. Advantages of Cylindrical Shapes

Many containers are cylindrical.

Examples:

Water tanks
Gas cylinders
Beverage cans

Reasons:

Cylinders can hold a large volume.
They distribute pressure evenly.
They are easy to manufacture.

Because of these advantages, cylindrical shapes are widely used.

77. Advantages of Spherical Shapes

Spheres are very efficient shapes.

Examples:

Football
Bubbles
Planets

Advantages:

Equal distribution of pressure.
Minimum surface area for a given volume.
Smooth rolling motion.

Because of these properties, spheres are common in nature and technology.

78. Advantages of Cuboid Shapes

Cuboids are commonly used for storage.

Examples:

Boxes
Rooms
Containers

Advantages:

Easy stacking
Efficient space usage
Stable structure

Cuboids can be arranged without leaving gaps.

79. Surface Area vs Volume

Students often confuse surface area and volume.

Surface Area

Surface area refers to the total area of the outer surface of a solid.

Example:

Wrapping paper needed for a box.

Volume

Volume refers to the space occupied inside the solid.

Example:

Amount of water a container can hold.

80. Relationship Between Size and Volume

When the size of a solid increases, its volume increases faster than its surface area.

Example:

If the side of a cube doubles:

Surface area increases 4 times.

Volume increases 8 times.

This concept is important in science and engineering.

81. Use of Three Dimensional Shapes in Packaging

Packaging industries use three dimensional shapes to pack products efficiently.

Examples:

Cuboid boxes for electronics
Cylindrical cans for drinks
Cone shaped ice cream packages

Packaging design considers:

volume
surface area
stacking ability

82. Mathematical Models of Solids

Mathematical models are used to represent real objects.

Example:

A ball can be represented as a sphere in mathematics.

These models help solve real world problems.

83. Visualization Skills

Visualization means imagining objects in three dimensional space.

Students should practice:

imagining shapes from different angles
drawing simple 3D diagrams
identifying shapes in everyday objects

These skills are important for higher mathematics.

84. Construction of Three Dimensional Shapes

Students can build models using:

cardboard
paper
clay
plastic sheets

Examples:

Make a cube using six squares.

Make a cylinder using a rectangle and two circles.

This helps understand the structure of solids.

85. Classroom Activities

Teachers may give activities such as:

Draw nets of different solids.
Build models of cubes and pyramids.
Identify 3D shapes in classroom objects.

These activities improve understanding.

86. Problem Solving Strategies

When solving questions related to three dimensional shapes:

Read the question carefully.
Identify the shape involved.
Write the correct formula.
Substitute values correctly.
Calculate step by step.

87. Word Problems

Example:

A cubical box has side 6 cm.

Find its volume.

Solution:

Volume = a³
= 6³
= 216 cm³

Example:

Find the volume of a cylindrical container.

r = 7 cm
h = 14 cm

Volume = πr²h
= 22/7 × 7² × 14
= 2156 cm³

88. Importance in Higher Classes

Three dimensional geometry becomes more advanced in higher classes.

Students later study:

mensuration
coordinate geometry
vectors
solid geometry

Understanding this chapter builds a strong foundation.

89. Common Exam Questions

Teachers often ask:

Define cube.
Define cuboid.
State Euler’s formula.
Write formulas for surface area.
Solve numerical problems.

Students should practice these questions regularly.

90. Final Revision Points

Important things to remember:

3D shapes have length, breadth and height.
Cube has 6 faces, 12 edges, 8 vertices.
Cuboid has rectangular faces.
Sphere has no edges or vertices.
Cylinder has two circular bases.
Cone has one circular base and a vertex.

Students must remember formulas for surface area and volume.

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Class 7 ICSE Mathematics
Three Dimensional Shapes – Advanced Extended Notes

Understanding Solid Geometry
The branch of mathematics that studies three dimensional objects is called solid geometry.
Solid geometry helps us understand objects that have:
length
breadth
height
These objects occupy space and therefore have volume.
Examples of solids:
Cube
Cuboid
Cylinder
Cone
Sphere
Prism
Pyramid
Solid geometry is useful in many real life fields such as engineering, architecture, and design.
Dimensions of Solids
Three dimensional shapes have three measurements.
Length
The longest side of an object.
Breadth (Width)
The side perpendicular to length.
Height (Depth)
The vertical measurement of the object.
Example:
A cuboid box has length, breadth, and height.
These three dimensions help determine volume and surface area.
Mathematical Representation of Solids
Mathematics uses symbols and formulas to represent solids.
Example:
Cube side = a
Volume of cube = a³
Surface area of cube = 6a²
Similarly, for a cuboid:
Length = l
Breadth = b
Height = h
Volume = l × b × h
Relationship Between Surface Area and Volume
Surface area and volume are closely related but represent different concepts.
Surface Area
The total area covering the outer surface of the object.
Example:
Paint required to cover a wall.
Volume
The space inside the object.
Example:
Water stored inside a tank.
Cubes in Everyday Life
Cubes appear in many everyday objects.
Examples:
Dice used in board games
Ice cubes
Sugar cubes
Puzzle cubes
These objects have equal sides and square faces.
Cuboids in Daily Life
Cuboids are one of the most common shapes in daily life.
Examples:
Books
Bricks
Boxes
Rooms
Furniture
Most storage containers are cuboid because they can be stacked easily.
Cylinders in Real Life
Cylinders are widely used in engineering and household items.
Examples:
Gas cylinders
Water tanks
Pipes
Batteries
Beverage cans
The circular base and curved surface make cylinders strong and efficient.
Cones in Real Life
Cones are used in many practical objects.
Examples:
Ice cream cones
Traffic cones
Party hats
Funnels
The pointed shape of cones helps in directing flow of liquids or materials.
Spheres in Real Life
Spheres are common in nature and sports.
Examples:
Balls used in sports
Planets
Soap bubbles
Marbles
Spheres roll easily and distribute pressure evenly.
Symmetry in Three Dimensional Shapes
Symmetry means balance and similarity of shape.
Many three dimensional shapes are symmetrical.
Example:
Cube
Each face is identical and the shape looks the same from many directions.
Sphere
A sphere has infinite symmetry because every point on the surface is equally distant from the center.
Symmetry is important in design and architecture.
Combination of Solids
Sometimes objects are made by combining two or more solids.
Example:
A toy may consist of:
a cylindrical body
a spherical head
In such cases, we calculate total volume or surface area by adding individual parts.
Subtraction of Solids
Sometimes part of a solid is removed.
Example:
A cylindrical hole drilled into a cube.
To calculate volume:
Volume of cube − volume of cylinder.
This concept is important in engineering and manufacturing.
Hollow and Solid Objects
Objects can be solid or hollow.
Solid Objects
Filled completely inside.
Example:
Stone cube.
Hollow Objects
Empty inside.
Example:
Pipe or hollow cylinder.
For hollow objects we may calculate:
inner surface area
outer surface area.
Measurement Accuracy
When measuring dimensions of objects, accuracy is important.
Tools used include:
ruler
measuring tape
vernier calipers
Accurate measurements ensure correct calculation of volume and surface area.
Scaling of Shapes
Scaling means increasing or decreasing the size of shapes.
Example:
If each side of a cube becomes twice as large:
Surface area becomes 4 times larger.
Volume becomes 8 times larger.
This concept is important in model making and architecture.
Three Dimensional Shapes in Computer Graphics
Modern technology uses three dimensional geometry.
Examples:
video games
animation movies
virtual reality
Computer graphics use mathematical models of solids to create realistic images.
Use in 3D Printing
3D printing technology creates objects layer by layer using digital models.
Engineers design objects using three dimensional geometry before printing them.
Examples:
prototypes
machine parts
medical devices.
Use in Robotics
Robots use sensors and cameras to detect objects in three dimensional space.
Understanding shapes helps robots:
identify objects
move safely
perform tasks accurately.
Environmental Applications
Three dimensional geometry is used in environmental studies.
Examples:
calculating water storage in dams
estimating volume of soil
measuring capacity of reservoirs.
Final Conclusion of the Chapter
Three dimensional shapes are solid objects that occupy space. They have three dimensions: length, breadth and height. Important solids include cube, cuboid, cylinder, cone, sphere, prism and pyramid.
Students learn about faces, edges and vertices of solids, and also study formulas for surface area and volume. These concepts are useful in everyday life as well as in fields such as architecture, engineering, science and technology.
Understanding three dimensional shapes improves spatial reasoning and helps students visualize objects around them.
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Class 7 ICSE Mathematics
Three Dimensional Shapes – Extended Study Notes (Continued)

Concept of Space in Geometry
Space is the region where objects exist and move. Three dimensional shapes exist in space because they occupy volume. When we study solid geometry, we are actually studying shapes that occupy space.
For example, a book kept on a table occupies space. The space inside a water bottle represents its capacity. Three dimensional shapes help us measure and understand such spaces.
Understanding space helps students imagine objects from different perspectives and improves spatial reasoning.
Interior and Exterior of a Solid
Every solid has two parts:
Interior
The inside region of the solid.
Example: The space inside a box.
Exterior
The outside region surrounding the solid.
Example: The air around the box.
The boundary between interior and exterior is the surface of the solid.
Boundary of a Solid
The outer covering of a solid is called its boundary.
For example:
The surface of a cube forms its boundary.
The curved surface of a sphere forms its boundary.
The boundary separates the inside of the solid from the outside environment.
Solid Objects Around Us
Many objects around us are examples of three dimensional shapes.
Examples include:
Object
Shape
Dice
Cube
Brick
Cuboid
Water pipe
Cylinder
Ice cream cone
Cone
Ball
Sphere
Recognizing these shapes helps students understand geometry better.
Nets of Three Dimensional Shapes
A net is a flat pattern that can be folded to make a solid.
When we cut a box and unfold it, we obtain its net.
Examples:
Cube → 6 squares
Cuboid → 6 rectangles
Cylinder → 1 rectangle + 2 circles
Nets help students understand the relationship between 2D and 3D shapes.
Folding and Unfolding of Solids
Folding and unfolding are important concepts in geometry.
Folding
Turning a flat shape into a solid.
Example: Folding paper squares to form a cube.
Unfolding
Opening a solid to form a flat net.
Example: Opening a cardboard box.
These activities improve spatial imagination.
Identifying Solids by Touch
Even without seeing an object, we can identify solids by touching them.
For example:
A ball feels round → sphere.
A dice has flat square faces → cube.
This shows how shapes can be understood through physical properties.
Counting Faces, Edges, and Vertices
Understanding faces, edges, and vertices is important.
Example: Cube
Faces = 6
Edges = 12
Vertices = 8
Example: Cuboid
Faces = 6
Edges = 12
Vertices = 8
Example: Cone
Faces = 2 (1 flat, 1 curved)
Edges = 1
Vertices = 1
Surface Area in Real Situations
Surface area calculations are used in many situations.
Examples:
Painting walls of a room
Designing packaging materials
Manufacturing containers
Surface area helps determine the amount of material needed.
Volume in Real Situations
Volume helps determine the capacity of containers.
Examples:
Water tanks
Fuel tanks
Storage containers
Volume calculations help measure how much material or liquid can be stored.
Estimating Volume
Sometimes exact measurement is not necessary.
We estimate volume using approximate measurements.
Example:
Estimating the volume of a storage box to know how many books it can hold.
Estimation is useful in practical situations.
Shapes Used in Sports Equipment
Many sports objects are based on three dimensional shapes.
Examples:
Football → sphere
Cricket ball → sphere
Basketball → sphere
Sports equipment is designed carefully using geometric principles.
Shapes Used in Transportation
Vehicles contain many three dimensional shapes.
Examples:
Wheels → cylinders
Fuel tanks → cylinders
Cargo containers → cuboids
Engineers design these shapes for efficiency and strength.
Shapes Used in Household Items
Many household items are examples of three dimensional shapes.
Examples:
Glass → cylinder
Bucket → cylinder
Cup → cylinder
Storage box → cuboid
These shapes are practical and easy to manufacture.
Shapes in Construction
Construction uses geometric solids.
Examples:
Buildings → cuboids
Water towers → cylinders
Roof tops → pyramids or cones
Engineers use geometry to ensure stability and safety.
Importance of Geometry in Modern Science
Geometry plays a major role in modern science.
Examples:
Astronomy studies planets and stars which are spherical.
Physics studies motion in three dimensional space.
Chemistry studies molecular structures in three dimensions.
Understanding geometry helps scientists explain natural phenomena.
Observation Skills
Students should observe shapes around them.
Examples:
Look at classroom objects and identify their shapes.
Identify shapes of buildings in your area.
Observe sports equipment.
Observation strengthens understanding of geometry.
Practical Activities for Students
Teachers may give practical activities such as:
Drawing nets of solids
Making paper models of cubes and pyramids
Identifying shapes in the classroom
These activities make learning more interesting.
Importance of Diagrams in Geometry
Drawing diagrams helps students understand shapes better.
Important diagrams include:
Cube
Cuboid
Cylinder
Cone
Sphere
Neat diagrams improve presentation in exams.
Final Revision Summary
Important points from the chapter:
Three dimensional shapes have length, breadth, and height.
Examples include cube, cuboid, sphere, cylinder, and cone.
Students learn about:
faces
edges
vertices
surface area
volume
These concepts help us understand objects around us and solve practical problems in daily life.
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Class 7 ICSE Mathematics
Three Dimensional Shapes – Extended Notes (Continued)

Understanding the Structure of Solids
Every three dimensional shape has a specific structure that determines its form and properties. The structure of a solid depends on how its faces, edges, and vertices are arranged.
For example:
A cube has six square faces arranged in such a way that they form a perfect solid shape. Each face meets another face at a straight line called an edge.
The point where three edges meet is called a vertex. The arrangement of these components determines the shape of the solid.
Comparing Different Solids
Different solids have different properties.
Shape
Faces
Edges
Vertices
Cube
6
12
8
Cuboid
6
12
8
Cylinder
3
2
0
Cone
2
1
1
Sphere
1
0
0
This comparison helps students understand how shapes differ from one another.
Similar Solids
Two solids are called similar if they have the same shape but different sizes.
Example:
Two cubes of different side lengths are similar.
The ratios of their corresponding sides remain constant.
Similarity is useful in designing models and scaled structures.
Congruent Solids
Two solids are called congruent if they have:
the same shape
the same size
Example:
Two identical dice are congruent cubes.
Congruent solids match exactly when placed on top of each other.
Scale Models
A scale model is a smaller or larger representation of an object.
Examples:
Model of a building
Model of a bridge
Model of a spacecraft
Architects and engineers use scale models to study designs before construction.
Surface Area of Rooms
Surface area calculations help determine materials needed for construction.
Example:
To paint a room, we calculate the area of its walls.
The room can be considered a cuboid.
Using surface area formulas, we determine the amount of paint required.
Volume of Storage Containers
Many storage containers are cuboids or cylinders.
Example:
A water tank may be cylindrical.
Volume formula:
Volume = πr²h
This formula helps calculate how much water the tank can store.
Efficiency of Shape Design
Engineers choose shapes carefully based on their efficiency.
Example:
Cylinders are used for storage tanks because they distribute pressure evenly.
Spheres are used for balls because they roll easily.
Cuboids are used for boxes because they stack efficiently.
Geometric Patterns in Nature
Nature often creates patterns based on geometry.
Examples:
Snowflakes have symmetrical shapes.
Crystals form geometric solids.
Honeycomb cells are hexagonal prisms.
These natural patterns show the importance of geometry in the natural world.
Geometry in Architecture
Architectural structures often use geometric shapes.
Example structures include pyramids, domes, and towers.
The Louvre Pyramid is a modern glass pyramid that uses geometric principles in its design.
Another famous structure with a spherical dome is the Taj Mahal, where domes resemble parts of spheres.
These examples show how geometry is used in real buildings.
Stability of Shapes
Some shapes are more stable than others.
Example:
Cuboids are stable because they have flat bases.
Cones are less stable because they have a pointed top.
Understanding stability helps engineers design safe structures.
Strength of Shapes
The strength of a structure depends on its shape.
Examples:
Triangles provide strong support in bridges.
Cylinders provide strength in pillars.
Domes provide strength in large buildings.
These shapes distribute forces effectively.
Three Dimensional Coordinates (Basic Idea)
Although studied in higher classes, the idea of three dimensional coordinates is simple.
Every point in space can be described by three numbers:
x-coordinate
y-coordinate
z-coordinate
These coordinates help locate points in three dimensional space.
Role of Geometry in Technology
Modern technology relies heavily on geometry.
Examples include:
Computer graphics
Engineering design
Architecture planning
Robotics
Three dimensional modelling is used to design products and structures.
Importance of Visualization Skills
Visualization helps students imagine shapes and objects in their mind.
Students should practice:
drawing 3D diagrams
identifying shapes in real life
constructing models
Visualization improves problem solving abilities.
Mathematical Thinking
Studying three dimensional shapes improves logical thinking.
Students learn to:
analyze shapes
apply formulas
solve practical problems
This helps develop mathematical reasoning skills.
Practice Problems
Question 1
Find the surface area of a cube with side 5 cm.
Solution:
Surface area = 6a²
= 6 × 5²
= 6 × 25
= 150 cm²
Question 2
Find the volume of a cuboid:
Length = 12 cm
Breadth = 7 cm
Height = 4 cm
Volume = 12 × 7 × 4
= 336 cm³
Question 3
Find the curved surface area of a cylinder.
Radius = 7 cm
Height = 10 cm
CSA = 2πrh
= 2 × 22/7 × 7 × 10
= 440 cm²
Word Problems
Example:
A cubical box has side 8 cm.
Find the volume of the box.
Solution:
Volume = a³
= 8³
= 512 cm³
Importance of Units
Units must always be written in answers.
Examples:
Surface area → square units (cm²)
Volume → cubic units (cm³)
Capacity → litres or millilitres
Correct units make answers meaningful.
Final Chapter Conclusion
Three dimensional shapes form an important part of geometry. These shapes include cube, cuboid, cylinder, cone, sphere, prism and pyramid. Each solid has unique properties such as faces, edges, vertices, surface area and volume.
Understanding these shapes helps students visualize objects around them and apply mathematical principles in real life. Three dimensional geometry is used in architecture, engineering, science, and technology.
Learning this chapter develops spatial thinking and prepares students for advanced mathematics in higher classes.
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Additional Notes – Three-Dimensional Shapes (Class 7 ICSE)

Difference Between Plane Shapes and Solid Shapes
Plane Shapes (2D)
Solid Shapes (3D)
Have length and breadth
Have length, breadth, and height
Flat figures
Solid objects
Example: square, rectangle, circle
Example: cube, cuboid, sphere
Example
A rectangle drawn on paper is a plane figure.
A box shaped like a cuboid is a solid figure.
Elements of Three-Dimensional Shapes
Every 3D shape has certain parts.
(a) Faces
A face is a flat surface of a solid.
Example
Cube → 6 faces
(b) Edges
The line where two faces meet is called an edge.
Example
Cube → 12 edges
(c) Vertices
A vertex is a corner where edges meet.
Example
Cube → 8 vertices
Euler’s Formula
Euler discovered a rule for polyhedra.
Formula:
F + V − E = 2
Where:
F = Number of faces
V = Number of vertices
E = Number of edges
Example (Cube)
F = 6
V = 8
E = 12
6 + 8 − 12 = 2 ✓
Polyhedra
A polyhedron is a solid with flat faces.
Examples:
Cube
Cuboid
Pyramid
Prism
Types of Polyhedra
Regular Polyhedron All faces are identical.
Example
Cube
Irregular Polyhedron Faces are not identical.
Example
Some pyramids
Prism
A prism is a solid with two parallel identical faces.
Examples:
Triangular prism
Rectangular prism
Properties of Prism
Two bases are identical
Side faces are rectangles
Cross section remains same
Example
A glass box is shaped like a prism.
Pyramid
A pyramid is a solid with:

One base
Triangular faces meeting at a point
Example
Egyptian pyramids
Types of Pyramid
Square pyramid
Triangular pyramid
Pentagonal pyramid

Net of 3D Shapes
A net is a flat pattern that can be folded to form a solid.
Examples
Cube Net
6 squares joined together.
Cuboid Net
6 rectangles joined together.
Cylinder Net

2 circles
1 rectangle

Cross Sections of Solids
When a solid is cut by a plane, the shape formed is called a cross section.
Examples
Cut a cylinder horizontally → Circle
Cut a cone horizontally → Circle
Cut a cube → Square or rectangle
Surface Area of Solids
Surface area means the total area of all faces of a solid.
Cube
Surface Area = 6a²
where a = side
Cuboid
Surface Area = 2(lb + bh + lh)
where
l = length
b = breadth
h = height
Volume of Solids
Volume means space occupied by a solid.
Cube
Volume = a³
Cuboid
Volume = l × b × h
Cylinder
Volume = πr²h
Difference Between Surface Area and Volume
Surface Area
Volume
Area of outer surface
Space inside the solid
Measured in square units
Measured in cubic units
Example cm²
Example cm³
Real Life Examples of 3D Shapes
Object
Shape
Dice
Cube
Book
Cuboid
Ball
Sphere
Ice cream cone
Cone
Water tank
Cylinder
Importance of Three-Dimensional Shapes
3D shapes are important in many fields:
Architecture Buildings and houses use 3D shapes.
Engineering Machines and tools use solid shapes.
Packaging Boxes are cuboids.
Sports Balls are spheres.
Applications in Daily Life
Examples:

Water bottles → Cylinders
Dice → Cube
Tent → Pyramid
Globe → Sphere
These shapes help us understand the structure of objects.

Visualisation of Solids
Visualisation means imagining the shape of objects in space.
Students can understand this by:

Looking at real objects
Drawing diagrams
Making models
Example
Imagine a cube made of matchsticks.

Symmetry in 3D Shapes
Some 3D shapes have symmetry.
Example
Cube → many planes of symmetry
Cylinder → circular symmetry
Symmetry helps in design and engineering.
Common Mistakes Students Make
Confusing edges and vertices
Forgetting units in volume
Mixing surface area and volume formulas
Drawing incorrect nets
Tips to Study This Chapter
Practice drawing shapes
Memorize formulas
Use real life objects for understanding
Solve many numerical problems
Practice Questions
Define a three-dimensional shape.
How many faces does a cube have?
What is the difference between prism and pyramid?
Write Euler’s formula.
Give three examples of 3D shapes from daily life.
Conclusion
The chapter Three-Dimensional Shapes introduces students to the world of solid figures. It explains the structure, properties, and formulas related to shapes like cubes, cuboids, cylinders, cones, and spheres. Understanding this chapter helps students develop spatial thinking and prepares them for advanced geometry in higher classes.
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More Detailed Notes – Three Dimensional Shapes (Class 7 ICSE)

Understanding Space and Solids
Three-dimensional shapes exist in space. Unlike flat figures drawn on paper, solids occupy a definite amount of space.
Space has three directions:
Length
Breadth
Height
Any object that has these three measurements is called a solid figure.
Examples:
A brick
A matchbox
A dice
All these are three-dimensional objects.
Types of Three-Dimensional Shapes
Three-dimensional shapes can be divided into two main groups.
(1) Polyhedra
These are solids that have flat faces.
Examples:
Cube
Cuboid
Prism
Pyramid
(2) Non-Polyhedra
These are solids that have curved surfaces.
Examples:
Cylinder
Cone
Sphere
Cube
A cube is a solid in which:
All faces are squares
All edges are equal
Properties of a Cube
Faces = 6
Edges = 12
Vertices = 8
Example
A dice used in games is a cube.
Surface Area of Cube
Surface Area = 6a²
Volume of Cube
Volume = a³
where a = length of edge
Cuboid
A cuboid is a solid whose faces are rectangles.
Examples:
Book
Brick
Matchbox
Properties of Cuboid
Faces = 6
Edges = 12
Vertices = 8
Opposite faces are equal and parallel.
Surface Area
Surface Area = 2(lb + bh + lh)
Volume
Volume = l × b × h
Cylinder
A cylinder has:
Two circular faces
One curved surface
Examples:
Water bottle
Pipe
Drum
Parts of Cylinder
Radius of base
Height
Curved surface
Curved Surface Area
2πrh
Volume
πr²h
Cone
A cone is a solid with:
One circular base
One curved surface
One vertex
Examples:
Ice cream cone
Party hat
Funnel
Parts of Cone

Base
Height
Slant height
Vertex

Sphere
A sphere is a perfectly round solid.
Examples:
Football
Globe
Orange
Properties

No edges
No vertices
Only one curved surface
All points on the surface are equidistant from the centre.

Hemisphere
A hemisphere is half of a sphere.
Example: Cutting a ball into two equal parts.
Examples:
Bowl
Dome
Nets of Solids
A net is the flat shape obtained when a solid is opened.
Example:
Cube net → six squares.
Cuboid net → rectangles.
Cylinder net →
2 circles + 1 rectangle.
Learning nets helps students understand how solids are formed.
Making Models of 3D Shapes
Students can create models using:

Paper
Cardboard
Clay
Matchsticks
For example:
Matchsticks + clay → cube model.
This activity improves spatial understanding.

Relationship Between Edges, Faces and Vertices
For polyhedra:
F + V − E = 2
Example for cuboid:
Faces = 6
Vertices = 8
Edges = 12
6 + 8 − 12 = 2
This rule always works for polyhedra.
Visualization of 3D Shapes
Visualization means imagining objects in space.
For example:
Imagine stacking small cubes to form a bigger cube.
Students can also imagine:

Cutting shapes
Rotating shapes
Folding nets
These skills are useful in geometry.

Stacking Cubes
Sometimes cubes are arranged together to form larger shapes.
Example:
Small cubes form a larger cube.
If each side has 3 cubes:
Total cubes = 3³ = 27
Projection of Solids
Projection means viewing the object from different directions.
Views of a solid:
Top view
Front view
Side view
Example:
A cube looks like a square from the top.
Cross Sections
When a solid is cut, the shape formed is called a cross section.
Examples:
Cube cut parallel to base → square.
Cylinder cut horizontally → circle.
Cone cut horizontally → circle.
Real Life Uses of 3D Geometry
Three-dimensional shapes are used in many areas.
Architecture
Buildings and bridges use cuboids and cylinders.
Engineering
Machines contain cylindrical and spherical parts.
Packaging
Boxes for products are cuboids.
Sports
Balls are spheres.
Fun Facts About 3D Shapes

The sphere has the smallest surface area for a given volume.
A cube has equal faces.
Pyramids were built thousands of years ago in Egypt.

Difference Between Prism and Pyramid
Prism
Pyramid
Two bases
One base
Side faces rectangles
Side faces triangles
Top face parallel to base
Faces meet at a point
Simple Numerical Examples
Example 1
Find the volume of a cube with side 5 cm.
Volume = a³
= 5³
= 125 cm³
Example 2
Find the volume of cuboid:
Length = 6 cm
Breadth = 4 cm
Height = 3 cm
Volume = l × b × h
= 6 × 4 × 3
= 72 cm³
Summary of the Chapter
In this chapter students learn:

Meaning of three-dimensional shapes
Different types of solids
Faces, edges, vertices
Nets of solids
Surface area and volume
Real life examples of solids
This chapter builds the foundation for solid geometry, which will be studied in higher classes.
If you want, I can also give:
50 ICSE exam-type questions from this chapter
Important formulas revision sheet
MCQs for Class 7 ICSE Three-Dimensional Shapes
A full 10,000-word ultra-detailed version for exams.

Here are even more extended notes for the Class 7 ICSE Mathematics chapter – Three-Dimensional Shapes to continue building very detailed study material.
Additional Detailed Notes – Three-Dimensional Shapes (Class 7 ICSE)

Understanding Solid Figures Through Observation
Students can easily understand three-dimensional shapes by observing objects around them.
Examples:
Object
Shape
Dice
Cube
Brick
Cuboid
Ice cream cone
Cone
Basketball
Sphere
Candle
Cylinder
Observing real objects helps students understand how geometry exists in everyday life.
Characteristics of Three-Dimensional Shapes
Every three-dimensional shape has certain important characteristics:
They occupy space.
They have volume.
They have surface area.
They can be viewed from different directions.
Example
A box has length, breadth, and height and therefore occupies space.
Surface Types in Solids
Surfaces in three-dimensional shapes can be of two types.
Flat Surfaces
These surfaces are completely plane.
Examples:
Cube
Cuboid
Pyramid
Curved Surfaces
These surfaces are not flat.
Examples:
Cylinder
Cone
Sphere
Combination of Solids
In real life, many objects are made by combining different solids.
Examples:

A house may have a cuboid base and pyramid roof.
A rocket may have a cylinder body and cone top.
This shows that real-world objects are often made using multiple geometric solids.

Understanding Vertices
A vertex is a corner point where edges meet.
Example:
In a cube, three edges meet at each vertex.
Total vertices in cube = 8
Vertices are very important when studying polyhedra.
Understanding Edges
Edges are the line segments where two faces meet.
Example:
Cube → 12 edges
Cuboid → 12 edges
Edges form the skeleton structure of the solid.
Understanding Faces
A face is a flat surface of a solid.
Example:
Cube → 6 square faces
Cuboid → 6 rectangular faces
Faces form the outer covering of the solid.
Drawing Three-Dimensional Shapes
Students often draw 3D shapes on paper.
Common methods:
Using perspective drawing
Using dotted lines for hidden edges
Example:
When drawing a cube, the back edges are shown with dotted lines.
Counting Cubes in a Structure
Sometimes cubes are stacked to form a structure.
Example:
If cubes are arranged in 2 layers, each layer containing 9 cubes:
Total cubes = 2 × 9
= 18 cubes
This type of problem helps develop visualization skills.
Hidden Cubes in a Solid
Sometimes cubes inside a structure cannot be seen.
Students must imagine the inside structure to count them correctly.
Example:
A cube made of 3 × 3 × 3 cubes
Total cubes = 27
But only some cubes are visible from outside.
Edge Length of Large Cubes
If small cubes combine to form a larger cube, we can calculate the edge length.
Example:
If 8 small cubes form a large cube:
Edge of large cube = 2 small cubes
because
2³ = 8
Painting Cubes Problems
A common problem involves painting a cube and cutting it into smaller cubes.
Students calculate:

Cubes with 3 painted faces
Cubes with 2 painted faces
Cubes with 1 painted face
Cubes with no paint
These problems improve logical thinking.

Understanding the Centre of a Solid
Many solids have a centre point.
Example:
Sphere → centre is equidistant from all points on surface.
Cube → centre lies in the middle of the solid.
Understanding centres helps in advanced geometry.
Rotation of Solids
Three-dimensional objects can rotate around an axis.
Example:
A globe rotates around its axis.
Similarly:
A cylinder can rotate around its central axis.
Rotation helps understand symmetry.
Symmetry in Solids
Symmetry means that a shape can be divided into identical parts.
Examples:
Cube → many planes of symmetry
Sphere → infinite symmetry
Cylinder → circular symmetry
Symmetry is important in art, architecture, and engineering.
Use of 3D Shapes in Architecture
Architects use 3D shapes when designing buildings.
Examples:

Cylinders for pillars
Cuboids for rooms
Cones for roofs
Domes shaped like hemispheres
Geometry helps architects design strong and beautiful structures.

Use of 3D Shapes in Engineering
Engineers use solid geometry when building machines.
Examples:

Pipes → cylinders
Bearings → spheres
Storage tanks → cylinders
Understanding shapes helps engineers design efficient machines.

Use in Computer Graphics
Modern computer graphics use 3D shapes.
Video games and animations create objects using:

Cubes
Cylinders
Spheres
This field is called 3D modeling.

Importance of Learning Three-Dimensional Shapes
Learning 3D shapes helps students:

Understand spatial relationships
Improve visualization skills
Prepare for higher geometry
Apply mathematics in real life
It also improves logical thinking and creativity.

Quick Revision Points
Students should remember:
3D shapes have length, breadth, and height.
Cube and cuboid are polyhedra.
Cylinder, cone, and sphere have curved surfaces.
Faces, edges, and vertices define polyhedra.
Volume measures the space occupied by a solid.
Final Conclusion
The chapter Three-Dimensional Shapes introduces students to the study of solid geometry. It helps them understand how objects occupy space and how different solids are structured. Students learn about cubes, cuboids, cylinders, cones, and spheres, along with their properties, faces, edges, and vertices.
Understanding these shapes builds a strong foundation for future mathematical concepts such as mensuration, geometry, and spatial reasoning.
If you want, I can also give:

100 ICSE exam-level questions from this chapter
Solved exercise questions
Important diagrams for the chapter
A full 15,000-word complete study guide for Class 7 ICSE Three-Dimensional Shapes.