Geometry & Measures
Grade 6-7Similar Shapes
Similar shapes are exactly the same shape but a different size, and GCSE questions test whether you can find a missing length using a scale factor, spot similar triangles hidden inside a larger diagram, and understand how a linear scale factor affects area and volume differently. This lesson covers finding a scale factor and missing length, similar triangles formed by a line parallel to one side, and how area and volume scale factors relate to the linear scale factor.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Scale factor and missing lengths
Two shapes are similar if one is an enlargement of the other, meaning all corresponding angles are equal and all corresponding sides are in the same ratio. To find a missing length, first find the scale factor by dividing a known length on the larger shape by the corresponding length on the smaller shape, then multiply (or divide) to find the missing length.
Triangle ABC is similar to triangle DEF, with AB = 5 cm and DE = 15 cm. The scale factor from ABC to DEF is 15 ÷ 5 = 3, so any length on DEF is 3 times the corresponding length on ABC.
Similar triangles from a parallel line
When a line inside a triangle is parallel to one of its sides, it creates a smaller triangle similar to the original, sharing the same angles. This is one of the most common ways similar triangles appear hidden inside a diagram.
In triangle ABE, DC is parallel to BE, with D on AB and C on AE. Since triangle ADC is similar to triangle ABE, the scale factor AB ÷ AD can be used to find BE from DC, or DC from BE.
Area scale factor
When two shapes are similar with linear scale factor n, their areas are in the ratio n², since area scales with both length and width. To go from a smaller area to a larger one, multiply by n²; to find the linear scale factor from a known area ratio, take the square root.
Two similar triangles have a linear scale factor of 3. If the smaller triangle has an area of 20 cm², the larger triangle has an area of 20 × 3² = 180 cm².
Volume scale factor
When two similar solids have linear scale factor n, their volumes are in the ratio n³, since volume scales with length, width and height. To find the linear scale factor from a known volume ratio, take the cube root.
Two similar solids have volumes of 27 cm³ and 216 cm³. The volume scale factor is 216 ÷ 27 = 8, so the linear scale factor is the cube root of 8, which is 2.
Worked Examples
Three exam-style questions, fully solved.
Triangle ABC is similar to triangle DEF. AB = 5 cm, BC = 7 cm, CA = 9 cm, DE = 15 cm, EF = 21 cm. Find the length FD.
Easy- 1.Find the scale factor using a pair of corresponding sides: 15 ÷ 5 = 3
- 2.Multiply the corresponding side CA by the scale factor: 9 × 3
Answer: 27 cm
Two similar triangles, ABC and DEF, have a linear scale factor of 3 (DE = 3 × AB). The area of triangle ABC is 20 cm². Find the area of triangle DEF.
Medium- 1.Find the area scale factor by squaring the linear scale factor: 3² = 9
- 2.Multiply the smaller area by the area scale factor: 20 × 9
Answer: 180 cm²
In triangle ABE, DC is parallel to BE, with D on AB and C on AE. AD = 6 cm, AB = 10 cm, DC = 9 cm, and the area of triangle ADC is 32.4 cm². Find (a) the length of BE, and (b) the area of triangle ABE.
Hard- 1.Find the linear scale factor: AB ÷ AD = 10 ÷ 6 = 5/3
- 2.Find BE by multiplying DC by the scale factor: 9 × 5/3 = 15 cm
- 3.Find the area scale factor by squaring the linear scale factor: (5/3)² = 25/9
- 4.Find the area of ABE by multiplying the area of ADC by the area scale factor: 32.4 × 25/9
Answer: BE = 15 cm, area of ABE = 90 cm²
Avoid These
The most common mistakes students make.
Using an area or volume ratio directly as a linear scale factor, instead of taking a square root (for area) or cube root (for volume) first.
Multiplying by the scale factor when a length should be divided by it, or the other way round, especially when going from the larger shape back to the smaller one.
Matching up the wrong pair of corresponding sides between two similar shapes, particularly when the shapes are drawn in different orientations.
Forgetting that a line parallel to one side of a triangle creates a smaller similar triangle, and instead treating the two triangles as unrelated.
Squaring or cubing the wrong scale factor, for example squaring a volume scale factor instead of finding the linear scale factor first.
FAQ
Questions parents and students ask.
Before this topic, make sure you know
What to learn next
Want a plan built around your child specifically?
Get our free 8-video course, or book a free Roadmap Call for a personalised plan.