Surface Area and Volume of Sphere and Hemispherical Sphere Volume – SS3 Mathematics Lesson Note (First Term)

A sphere is a solid in which all points on the round surface are the same distance from a fixed point, the centre. That distance is the radius. The volume of a sphere is four-thirds of pi times the cube of the radius, and its surface area is 4πr². A hemisphere is half a sphere, and its volume is two-thirds of πr³.

Behavioural objectives

By the end of the lesson, students should be able to:

• define a sphere and a hemisphere;
• state the radius and diameter of a sphere and the link between them;
• state the formulae for the surface area and volume of a sphere;
• state the formulae for the curved surface area, total surface area and volume of a hemisphere;
• solve problems on the volume of a sphere, including finding the radius from a given volume.

What is mensuration?

Mensuration is the branch of Mathematics that deals with measurement, especially the derivation and use of algebraic formulae to measure the areas, volumes and other parameters of geometric shapes. Examples of shapes covered are the cylinder, cone, cuboid, rectangular prism and rectangular-based pyramid, including the total surface area and volume of the cylinder and cone.

What are the formulae for common solids?

A sphere

A hemisphere

What is a sphere?

A sphere is a solid in which all the points on the round surface are the same distance from a fixed point, known as the centre of the sphere. The distance from the centre to the surface is the radius. The longest straight distance through the centre of a sphere is the diameter. The diameter is twice the radius.

How do you find the volume and surface area of a sphere?

The volume of a sphere is equal to four-thirds of the product of pi and the cube of the radius.

The volume and surface area of a sphere are given by these formulae, where r is the radius of the sphere:

• Volume, V = (4/3)πr³
• Surface area, A = 4πr²

We can also change the subject of the formula to obtain the radius when the volume is given.

Example 1

Calculate the volume of the sphere with radius 4 cm.

Solution:

Volume of sphere = (4/3)πr³

Example 2

The volume of a spherical ball is 5,000 cm³. What is the radius of the ball?

Example 3

Find the volume of a sphere with a diameter of 14 cm.

What is the volume of a hemisphere?

A hemisphere is half a sphere, with one flat circular face and one bowl-shaped face.

The volume of a hemisphere = (2/3)πr³, where r is the radius.

Evaluation

1. Define a sphere.
2. What is the relationship between the radius and the diameter of a sphere?
3. Write down the formulae for the volume and the surface area of a sphere.
4. Write down the formulae for the curved surface area, total surface area and volume of a hemisphere.
5. Calculate the volume of a sphere with radius 3 cm. Leave your answer in terms of π.

Assignment

1. Calculate the surface area of a sphere with radius 6 cm. Leave your answer in terms of π.
2. Find the volume of a hemisphere with radius 7 cm. (Take π = 22/7.)
3. Find the volume of a sphere with a diameter of 10 cm. Leave your answer in terms of π.