Similarity in Shapes

Triangle ADE sits inside triangle ABC, with line DE parallel to line BC. The length of AD is 4 cm, the length of DB is 8 cm, and the length of DE is 5 cm. Calculate the length of BC.

Step 1: Separate the overlapping triangles and identify corresponding sides.

• Small triangle $ADE$ has side $AD = 4$ cm.
• Large triangle $ABC$ has corresponding side $AB = 4 + 8 = 12$ cm.

Step 2: Calculate the linear scale factor ($k$).

• $k = 12 \div 4 = 3$

Step 3: Apply the scale factor to find the missing length.

• The side $DE$ ($5$ cm) corresponds to $BC$.
• $BC = 5 \times 3$

Step 4: State the final answer with units.

• $BC = 15$ cm

Understanding the Length, Area, and Volume Relationship

The relationship between the dimensions of similar shapes relies heavily on the dimension you are measuring. A common question is why we square the scale factor for area and cube it for volume.

Imagine a simple unit cube with a side length of $1$ cm. Its front face has an area of $1 \times 1 = 1$ cm$^2$, and its volume is $1 \times 1 \times 1 = 1$ cm.

If we enlarge this cube by a linear scale factor of $k = 4$, the new side length becomes $4$ cm. Because area is two-dimensional (length $\times$ width), the new face area becomes $4 \times 4 = 16$ cm$$. Notice that is exactly . Because volume is three-dimensional (length width height), the new volume becomes cm. Notice that is exactly .

Comparing Scale Factors

The table below summarises how scaling applies across different mathematical dimensions, demonstrating the clear link between the linear scale factor ($k$), area scale factor ($k^2$), and volume scale factor ($k^3$).

Feature	Linear	Area	Volume
What it measures	1D (Lengths, perimeters, radii)	2D (Surface areas, cross-sections)	3D (Capacity, volumes, masses of constant density)
Scale Factor	$k$	$k^2$	$k^3$
Ratio Format	$$

Working Backwards

You can easily move backwards from area or volume to find missing lengths. If you are given the ratio of two areas, you must calculate the square root of the area scale factor ($\sqrt{k^2}$) to find the linear scale factor ($k$). If you are given two volumes, you must calculate the cube root of the volume scale factor ($$) to find the linear scale factor (). Once you have found , you can scale any length on the shape.