Sheet № 156 · Foundation + Higher · AQA · Edexcel · OCR
Similar Shapes and Scale Factors –
Similar shapes have the same shape but different sizes — all corresponding angles are equal and corresponding sides are in the same ratio. This topic spans Foundation and Higher GCSE Maths, with Higher students needing to handle area and volume scale factors. It appears regularly on AQA, Edexcel, and OCR exams. This guide explains linear,
§Key definitions
Question:
Triangles A and B are similar. Triangle A has sides 4 cm, 6 cm, and 8 cm. The shortest side of triangle B is 10 cm. Find the other two sides of triangle B.
Answer:
The sides of triangle B are 10 cm, 15 cm, and 20 cm.
Q1 (Foundation):
Two similar rectangles have widths 3 cm and 9 cm. The length of the smaller rectangle is 5 cm. Find the length of the larger rectangle.
Q2 (Foundation):
Two similar triangles have corresponding sides of 8 cm and 12 cm. Find the linear scale factor from the smaller to the larger.
Q3 (Higher):
Two similar prisms have surface areas of 50 cm² and 200 cm². The volume of the smaller prism is 80 cm³. Find the volume of the larger prism.
§Formulas to memorise
Linear scale factor (k) = new length ÷ original length
Area scale factor = k²
Volume scale factor = k³
Identify corresponding sides — by matching angles or using the diagram.
Calculate the linear scale factor — by dividing a known side of one shape by the corresponding side of the other.
Find missing lengths — by multiplying or dividing by k.
For area problems — (Higher), square the linear scale factor to get the area scale factor.
For volume problems — (Higher), cube the linear scale factor to get the volume scale factor.
Worked example
Triangles A and B are similar. Triangle A has sides 4 cm, 6 cm, and 8 cm. The shortest side of triangle B is 10 cm. Find the other two sides of triangle B.
Working:
⚠ Common mistakes
- ✗Using the linear scale factor for area or volume. If k = 3, the area scale factor is 9 (not 3) and the volume scale factor is 27 (not 3). Students frequently forget to square or cube.
- ✗Matching the wrong sides. Always pair the shortest side with the shortest side, or use the diagram to identify corresponding sides by their position relative to equal angles.
- ✗Dividing instead of multiplying (or vice versa). If going from small to large, multiply by k. If going from large to small, divide by k.
✦ Exam tips
- →Write down the scale factor clearly before doing any further calculations — this avoids errors in multi-step problems.
- →For area and volume problems, it can help to write: "Linear SF = ..., Area SF = (...)², Volume SF = (...)³" as a checklist.
- →If the question gives you areas, find the area scale factor first, then square root it to get the linear scale factor.
- →If the question gives you volumes, find the volume scale factor first, then cube root it to get the linear scale factor.
- →Real-world contexts (maps, models, packaging) often appear — the same rules apply.