Area & Volume of Similar ShapesEdexcel IGCSE Maths: Revision notes

Section 1

What does 'similar' mean here?

Two shapes are similar if one is an enlargement of the other — same shape, different size. Every pair of corresponding lengths is in the same ratio, called the linear scale factor, (k).

If shape B is an enlargement of shape A with linear scale factor (k): $k = \frac{\text{length on B}}{\text{corresponding length on A}}$

This applies to any corresponding length: sides, diagonals, heights, radii, perimeters.

Key termssimilar shapeslinear scale factor

Exam tip

Always find (k) using a pair of matching (corresponding) lengths — check the shapes are labelled consistently before dividing.

Section 2

Why does area scale by (k^2)?

Area is a two-dimensional measurement — it comes from multiplying two lengths together (e.g. length × width, or $\pi r^2$ ). If every length is multiplied by $k$ , then area is multiplied by $k \times k = k^2$ .

$\text{Area scale factor} = k^2$

So if the area of shape A is $A_1$ and shape B is similar with linear scale factor $k$ : $A_2 = A_1 \times k^2$

This works for any similar 2D shapes — triangles, circles, rectangles, composite shapes — not just simple ones.

Key termsarea scale factor

Think of it like this

Think of tiling a floor: double every side length of a tile ((k=2)) and you need (2^2 = 4) times as many original tiles to cover the same shape scaled up.

Example

A triangle has area 12 cm $^2$ . A similar triangle has sides 3 times as long ( $k=3$ ). New area = $12 \times 3^2 = 12 \times 9 = 108$ cm $^2$ .

Section 3

Why does volume scale by (k^3)?

Volume is a three-dimensional measurement — it comes from multiplying three lengths (e.g. length × width × height, or $\frac{4}{3}\pi r^3$ ). If every length is multiplied by $k$ , volume is multiplied by $k \times k \times k = k^3$ .

$\text{Volume scale factor} = k^3$

If the volume of solid A is $V_1$ and solid B is similar with linear scale factor $k$ : $V_2 = V_1 \times k^3$

This also means mass scales by $k^3$ for solids made of the same material (same density), since mass = density × volume.

Key termsvolume scale factor

Think of it like this

Doubling every dimension of a cube ((k=2)) turns 1 small cube into a block you could pack with (2^3 = 8) of the original cubes.

Example

A model car has volume 20 cm $^3$ . The real car is similar with linear scale factor 15. Real volume = $20 \times 15^3 = 20 \times 3375 = 67{,}500$ cm $^3$ .

Section 4

How do I work backwards from area or volume to find (k)?

Exam questions often give you the area or volume ratio and ask for the linear scale factor, or ask for a missing length using that ratio.

If you know the area scale factor, take the square root to get (k): $k = \sqrt{\text{area scale factor}}$
If you know the volume scale factor, take the cube root to get (k): $k = \sqrt[3]{\text{volume scale factor}}$

Worked method for a length problem:

Find the area (or volume) ratio between the two shapes.
Square-root (area) or cube-root (volume) to get (k).
Multiply or divide the known length by (k) to find the missing length.

Example

Two similar cylinders have volumes 64 cm $^3$ and 216 cm $^3$ . Volume ratio = $216 \div 64 = 3.375$ . $k = \sqrt[3]{3.375} = 1.5$ . If the small cylinder's height is 4 cm, the large one's height is $4 \times 1.5 = 6$ cm.

Common mistake

Do not square-root or cube-root a length by mistake, and never apply (k^2) or (k^3) to a length — only areas use (k^2) and only volumes use (k^3).

Section 5

Where do students lose marks?

The three scale factors ((k), (k^2), (k^3)) are easy to mix up under exam pressure, especially when a question gives an area or volume ratio rather than a length ratio.

Common mistake

Using (k) instead of (k^2) when scaling an area, or (k^2) instead of (k^3) when scaling a volume — always check what quantity (length, area, volume) you are scaling before choosing the power.

Common mistake

Finding the scale factor the wrong way round (small ÷ large instead of large ÷ small), which flips the ratio upside down.

Common mistake

Forgetting to root the ratio when going from area/volume back to a length — students often multiply by (k^2) or (k^3) again instead of square/cube rooting.

Exam tip

Write (k), (k^2), (k^3) at the start of your working and label which one applies to lengths, areas and volumes — this stops mix-ups mid-calculation.

Must Know

Linear scale factor (k) = ratio of corresponding lengths.
Area scale factor = (k^2); Volume scale factor = (k^3).
To go from area ratio back to (k), take the square root; from volume ratio, take the cube root.
These rules apply to any similar shapes/solids, not just cubes or spheres.
Same-material solids: mass and volume both scale by (k^3).
Always double-check whether the question is about a length, an area, or a volume before applying (k), (k^2), or (k^3).

That's the notes covered.

Carry on to the next subtopic.