Volume of Similar Shapes

April 3, 2023

Differentiated Learning Objectives

All students should calculate the volume scale factor given two corresponding lengths in similar solids.
Most students should work out and apply the volume scale factor to calculate unknown volumes in similar solids.
Some students should work out and apply the volume and length scale factors to calculate unknown measurements in solid shapes.

Links to Lesson Resources (Members Only)

Starter/Introduction

Students recap the length and area scale factor using two similar pentagons. For example, to work out the area of the larger pentagon marked x, encourage students to begin by finding the length scale factor using a matching pair of sides. From this, they can work out the area scale factor using the relationship Area_S.F.= Length_S.F.².

Students should clearly show how they worked out the area scale factor as a similar approach will be needed later in the lesson when finding the volume scale factor.

Prompts / Questions to consider

Which pair of corresponding lengths can be used to work out the length scale factor?
What is the relationship between the length and area scale factors.

Volume of Similar Shapes

In the central part of the lesson, present the three cubes on the board and ask students to work in pairs to consider any relationship between the cube lengths, surface areas and volumes.

After a few minutes of processing time and small group discussions, present the table below and ask students to complete it using their ideas so far.

Solution

Using the table above as a guide, students should now be able to define the relationship between the length and volume scale factor as Volume_S.F.= Length_S.F.³

At this point, the teacher could work through questions 1 and 2 on the second slide as shown in the video above with the class attempting Q3 on mini-whiteboards.

Prompts / Questions to consider

Which scale factor do we need to work out an unknown volume.
What is the relationship between the length, area and volume scale factors?
Is it easier to leave non integer scale factors as decimals or simplified fractions? Why?

Plenary

The plenary challenges students to work out the volume of a frustum by considering it as two similar square-based pyramids. Students could prove that the two pyramids are mathematically similar using angles in parallel lines.

All students should include clear diagrams as part of their work as this will help them identify how to use the relevant scale factors.

Prompts / Questions to consider

How do we know the smaller pyramid at the top is mathematically similar to the larger blue pyramid?
Which pair of sides give us the length scale factor.
How is the larger blue pyramid changed to make the frustum?

Differentiation

More able students could work with shapes involving fractional scale factors and consider the relationship between the volume and area scale factors. Less able students may benefit from working with solids with integer scale factors.