# Lesson Notes By Weeks and Term - Junior Secondary School 1

THREE DIMENSIONAL SHAPES

SUBJECT: MATHEMATICS

CLASS:� JSS 1

DATE:

TERM: 3rd TERM

REFERENCE TEXTBOOKS�

• New General Mathematics, Junior Secondary School Book 1
• Essential Mathematics for Junior Secondary School Book 1

WEEK FIVE

TOPIC: THREE DIMENSIONAL SHAPES

Three dimensional ( 3-D) shapes are also called solid shapes. They have length, breadth and height unlike 2-D shapes that have only length and breadth. Examples of 3-D shapes are cubes, cuboids, cylinders, prisms, pyramids and spheres. They are also called geometrical solids.

Key words

Face: a surface of solid shape

Edge: a line on a solid where two faces meet

Vertex (plural vertices): a point or corner on a solid, usually where edges meet

Net: a flat shape that you can fold to make a solid

Cuboids and Cubes

• A Cube

���

����������������������������������������������������������������������������������������������������������������������������������������������������������������Net

A cube has the following properties

1. It has 12 straight edges
2. It has 8 vertices
3. It has 6 square faces
4. Its net consist of 6 square faces joined together

• A cuboid

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Net of cuboid�

A cuboid has the following properties:

1. It has 12 straight edges
2. It has 8 vertices
3. It also has 6 rectangular faces
4. Its net consist of 6 rectangular faces

Cylinders and Prisms

A Cylinder

Properties:

1. A cylinder has two circular faces
2. It has 1 curved surface
3. It has 2 curved edges
4. Its net consists of two circular faces and 1 rectangular face i.e. its net consist of 2 circles and 1 rectangle.

The net of a cylinder has two circles and one rectangle

Prism

The base and top faces of a prism are always the same shape. The names of prisms come from the shape of their base and top faces.

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���������������������������������Triangular Prism� � � � � � � � Hexagonal prism

Cones and pyramids

Cone

A cone is a solid shape with curved body, circular base and a pointed end.

Pyramid

A pyramid is a solid shape with a flat base and triangular faces rising to meet at a common point called its vertex. There are many types of pyramid. The different types are named after the shapes of the bases they have:

������������������������������������������������������������������������Rectangle pyramid � � � � � � � � � � � � � � Trapezoid Pyramid

Sphere

A sphere is a solid shape with perfectly round surface. Examples are orange, ball, shotput, etc.

Volumes of Solids

Volume of Cuboids

The volume of solids is a measure of the amount of space it occupies. A solid object is also called a 3-dimensional ( 3-D) object. The cube is used as the basic shape to estimate the volume of solid. Therefore, volume is measured in cubic unit.� A cube of an edge 1cm has a volume of one cubic centimetre (1cm3).

The volume of a cuboid is given by:

Volume= length x width x height� � � � � � � i.e.� V = l x w x h

In the above formula, A = l x w� where A= base area of the cuboid

Hence: Volume of a cuboid = base area x height

V = A x h

Volumes of cubes

When all the edges of a cuboid are equal, it is called a cube. If one edge is l unit long, then�

Volume of a cube = length x height x width

i.e � V = l�l�l

����������= l3

A cube of an edge 3cm will have a volume of 3 x 3 x 3 = 27cm3.

The above formula can be used to find the edge of a cube when the volume is given.

l3 = V

l = 3V

Example 1

Calculate the volume of a rectangular tank with dimensions 20cm by 15cm by 12 cm.

Solution

Volume = length x width x height

V = l x w x h

��= (20 x 15 x 12) cm3

��= 3600cm3

Example 2

A cuboid, 12 cm long and 8cm wide has a volume of 624cm3. Find the height of the cuboid.

Solution�

��������������������������������V = 624cm3

Substituting V = 624cm3, l = 12cm, and w = 8cm

Length x width x height = volume

L x w x h = V

12 x 8 x h = 624

96h = 624

Divide both sides by 96,� h = 62496 = 6.5cm

The height of the cuboid = 6.5cm

Example 3

A tank of water in the shape of a cuboid has a square base. If the depth of water in the tank is 3m high and the volume of the water inside the cuboid is 243m2. Calculate the width of the tank.

Solution

Volume of a cuboid= base area x height

Since it has a square base, the base area = l2 � , i.e. l = w.

243m3 = l2 x 3m

l2= 243m23cm = 81m2

Therefore, l = 281 = 9m

The width of the tank is 9m

Evaluation:

1. A cube volume of a cube is given as 512cm3
1. What is the length of one edge of the cube?
2. How many small cubes of edge 2cm can be placed together to make this cube?
1. A cuboid has a base area of 35cm2 and a height of 3.5cm. What is the volume of the cuboid?

General Evaluation/Revision Questions

1. A rectangular prism ( cuboid) has a volume of 680cm3 and its height is 20cm. What is the area of the base of the prism?
2. The base of a swimming pool is 192m2. The depth of the swimming pool is 1.8m. find the volume of water the swimming pool can hold.
3. A book measures 18cm by 12cm by 3cm. Calculate its volume

• Essential Mathematics for J.S.S 1 by A. J. S Oluwasanmi, page 212-222.�

Weekend Assignment

1. What is the volume of a cube of edge 5cm. (a) 15cm3� (b) 75cm3 � (c) 125cm3 � (d)� 25cm3
2. Find the volume of air in a container whose dimensions are: length = 25cm, width = 20cm and height= 10cm (a) 5000cm3� (b) 2500cm3 (c) 4500cm3 � (d) 500cm3
3. The volume of a cube is given as 512cm3. What is the length of one edge of the cube?� (a) 10cm� (b) 6cm � (c)� 4cm� (d) 8cm
4. How many small cubes of edge 2cm can be placed together to make the cube in question 3 above? (a) 66 (b) 32� (c) 64 � (d) 128
5. Calculate the volume of a cuboid with dimension 18cm by 12cm by 8cm. (a)� 1728cm3� (b)� 512cm3� (c)� 144cm3� (d)� 1872cm3

Theory

1. The base of a cuboid has one side equal to 10cm, and the other side is 5cm longer. If the height of the cuboid is 7cm, find the volume of the cuboid.
2. A cuboid measures xcm by 3xcm by 5xcm
1. Work out the volume of the cuboid in terms of x
2. What is the volume of the cuboid if x = 10cm?