Higher – Similarity in 2D and 3D shapes

Calculating lengths, areas and volumes.

Let's look at the relationship between length, area and volume.

This cube has side lengths of 1 unit.

That means the surface area of each one of its faces is 1 multiplied by 1, which is 1 square unit, and its volume is 1 multiplied by 1 multiplied by 1, which is 1 cubic unit.

What about this cube which has side lengths of 2 units?

This time, the area of each face is 2 multiplied by 2, which is 4 square units, and the volume is 2 multiplied by 2 multiplied by 2, which is 8 cubic units.

Notice that the side length has increased by a factor of 2, the area has increased by a factor of 4, or 2 squared, and the volume has increased by a factor of 8, or 2 cubed.

If instead you increase the side length by a factor of 3, this means the area will increase by a factor of 9, or 3 squared, and the volume will increase by a factor of 27, or 3 cubed.

In fact, if each side length of a 3D shape is increased by a scale factor of 𝑥, then the area of each face would increase by a scale factor of 𝑥 squared, and the volume of the shape would increase by a scale factor of 𝑥 cubed.

For example, shapes A and B are similar.

Question part a: find the missing lengths 𝑦 and 𝑧.

‘Similar’ means that although the two shapes are not the same size, their side lengths are in the same ratio.

In other words, shape B is the same shape as A, but a certain number of times bigger.

To find 𝑦 and 𝑧, you need to find a pair of corresponding lengths which you can use to calculate the length scale factor.

Corresponding sides in similar shapes have lengths that are in proportion to one another, and the angles at the vertices between the sides are the same in both shapes.

The longest side in shape A is 1 centimetre and the longest side in shape B is 4 centimetres, so the length scale factor is 4 divided by 1, which equals 4.

So, the side lengths in shape B are all 4 times bigger than the corresponding side lengths in shape A.

In other words, to get from lengths in shape A to lengths in shape B, you multiply by 4, and to get from lengths in shape B to lengths in shape A, you divide by 4.

Now use the length scale factor to find the missing lengths 𝑦 and 𝑧.

Starting with 𝑦, the corresponding side in shape B is 2 centimetres, so 𝑦 equals 2 divided by 4, which is 0.5 centimetres.

Likewise, for 𝑧, the corresponding side in shape A is 0.7 centimetres, so 𝑧 equals 0.7 multiplied by 4, which is 2.8 centimetres.

Now, shape A has a surface area of 1.9 centimetres squared and a volume of 0.12 centimetres cubed.

Question part b: find the surface area and volume of shape B.

Since the length scale factor is 4, the area scale factor is 4 squared, or 16, and the volume scale factor is 4 cubed, or 64.

This means the surface area of shape B is 1.9 multiplied by 16, which equals 30.4 centimetres squared, and the volume is 0.12 multiplied by 64, which equals 7.68 centimetres cubed.