What you'll learn
This topic covers similarity — identifying similar shapes and using scale factors for lengths, areas and volumes. In this guide you will learn what makes shapes similar, how to find and use a length scale factor, how area and volume scale factors differ, and how to solve problems with similar triangles. These are important geometry and reasoning skills.
Key terms and definitions
Similar — the same shape with equal angles and sides in the same ratio.
Scale factor — the number you multiply lengths by to enlarge a shape.
Length scale factor (k) — the ratio of corresponding lengths.
Area scale factor — k², the ratio of corresponding areas.
Volume scale factor — k³, the ratio of corresponding volumes.
Core concepts
What makes shapes similar
Two shapes are similar if they have equal corresponding angles and corresponding sides in the same ratio. They are the same shape but possibly different sizes — unlike congruent shapes, which are identical. Similar triangles are especially common.
Finding the length scale factor
The length scale factor k is found by dividing a length on one shape by the corresponding length on the other. Once known, multiply other lengths by k (or divide, going the other way) to find missing sides.
Area scale factor
Areas scale by the square of the length scale factor: area scale factor = k². If lengths are doubled (k = 2), areas become 4 times bigger. This is a frequent source of error — never use k for areas.
Volume scale factor
Volumes scale by the cube of the length scale factor: volume scale factor = k³. If lengths are doubled, volumes become 8 times bigger. So given a volume ratio, take the cube root to get the length scale factor.
Similar triangles in problems
In similar triangles, matching sides are in proportion. Identify corresponding sides (often using equal angles or parallel lines), set up the ratio, and solve for the unknown. Parallel lines inside a triangle create similar triangles.
Worked examples
Example 1: Missing length
Two similar shapes have corresponding sides 4 cm and 10 cm. A 6 cm side on the small shape corresponds to what?
k = 10 ÷ 4 = 2.5, so 6 × 2.5 = 15 cm.
Example 2: Area scale factor
Lengths of two similar shapes are in ratio 1 : 3. Find the ratio of areas.
Area ratio = 3² = 1 : 9.
Example 3: Volume scale factor
Two similar solids have volumes in ratio 1 : 27. Find the length ratio.
Cube root of 27 = 3, so 1 : 3.
Common mistakes and how to avoid them
Using k for areas or volumes. Use k² for area and k³ for volume.
Matching the wrong sides. Pair up corresponding sides using equal angles.
Forgetting to take roots. From an area ratio take the square root; from a volume ratio, the cube root.
Confusing similar and congruent. Similar can differ in size.
Inverting the scale factor. Check whether you are enlarging or reducing.
Exam technique for Similarity
Check equal angles and proportional sides for similarity.
Find the length scale factor from corresponding sides.
Use k² for area and k³ for volume.
Take square or cube roots to go back to lengths.
Identify corresponding sides carefully in similar triangles.
Quick revision summary
Two shapes are similar if their angles are equal and corresponding sides are in the same ratio — the same shape at a possibly different size. The length scale factor k comes from dividing corresponding lengths; multiply by k to find missing sides. Crucially, areas scale by k² and volumes scale by k³: doubling lengths (k = 2) makes areas 4× and volumes 8× bigger. Working backwards, take the square root of an area ratio or the cube root of a volume ratio to recover the length scale factor. In similar triangles (often created by parallel lines), pair corresponding sides using equal angles, set up the ratio and solve. The common errors are using k instead of k² or k³, matching the wrong sides, forgetting to take roots, and confusing similar with congruent. Check angles and ratios, find k, use k² and k³ for area and volume, and pair corresponding sides carefully.