Form 4 & 5 Unit 7 Lesson 3 – Volumes of Similar Objects

Objective

At the end of this lesson, students should be able to:

• Use the relationship between volumes of similar shapes with corresponding results for similar solids.

When objects are similar, one is an accurate enlargement of the other. If two objects are similar and the ratio of corresponding sides is k, then the ratio of their volumes is k³.

Earlier in the chapter, we found the following relationship between the scale factor and the area of factor of enlargement.

Area factor = (scale factor)²

An area has two dimensions, and the scale factor is used twice.

A similar relationship can be stated for volumes of similar shapes.

Volume factor = (scale factor)³

A volume has three dimensions and the scale factor is used three times.

Example 1

Two similar cylinders have height 4cm and 8cm respectively. If the volume of the smaller cylinder is 25cm³, find the volume of the larger cylinder.

Solution

Example 2

Two similar cones made of the same material have masses of 32kg and 108kg respectively. If the radius of the larger cone is 9cm, find the radius of the smaller cone.

Solution

Unit 7 Lesson 3: Exercise 1

1) The area of the bases of two similar glasses is in the ratio 4:25. Find the ratio of their volumes.

2) Find the volume (V) of the following similar objects.

3. Find the lengths marked by a letter.

4) A water container of height 2.5m has volume of 300m³. What is the volume of a similar container 7.5m high?

5) Two similar cylinders A and B can hold 160cm³ and 540cm³ of liquid respectively. If the radius of A is 12cm, determine the radius of B.

NOTE: Please check for the Answers in the (Additional) Material section at the top

Unit 7 Lesson 3: Exercise 2

1. Two similar prisms A and B are of heights 8cm and 10cm respectively. If A can hold 832cm³ of liquid, find the capacity of B.
2. The wood for packing case is 50cm long and has a volume of 80,000cm³. Find the volume of a similar wood if the length is increased by 10cm.
3. The masses of two similar objects are 24kg and 81kg respectively. If the surface area of the larger object is 540cm², find the surface area of the smaller object.
4. Two solid spheres have surface areas of 5cm² and 45cm² respectively and the mass of the smaller sphere is 2kg. Find the mass of the larger sphere.
5. Two similar jugs have heights of 4cm and 6cm respectively. If the capacity of the smaller jug is 50cm³, find the capacity of the larger jug.

NOTE: Please check for the Answers in the (Additional) Material section at the top