Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Geometry and measures
Practise Similar shapes (length, area, volume). unlimited generated questions on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Similar shapes are the same shape but a different size: their angles are equal and all their lengths are in the same ratio. You use the scale factor to find missing lengths and, on Higher, areas and volumes of similar shapes and solids.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
4
Recognise similar shapes and find scale factorsCheck that corresponding sides are all in the same ratio; scale factor = new length ÷ original length.
5
Find missing lengths in similar shapesMultiply or divide a length by the scale factor.
5
Use similar triangles made by parallel linesParallel lines give equal corresponding or alternate angles, so the triangles are similar.
Notes
Similar shapes
Two shapes are similar if one is an enlargement of the other: corresponding angles are equal and corresponding sides are in the same ratio.
Length scale factor \(k = \dfrac{\text{length on the new shape}}{\text{corresponding length on the original}}\).
Match corresponding sides carefully. If one shape is turned or flipped, redraw them the same way round.
Two triangles are similar if their angles are equal. Two pairs of equal angles is enough, as the third angles must then be equal too.
Missing lengths
Find k from a pair of corresponding sides you know. Multiply by k to go from the small shape to the large one; divide by k to go back.
A line parallel to one side of a triangle cuts off a smaller, similar triangle (the corresponding angles are equal).
Two parallel lines crossed by two straight lines that meet between them make an 'hourglass' of two similar triangles (alternate angles and vertically opposite angles are equal).
Use whole sides: if AD = 4 cm and DB = 6 cm on the same line, the full side AB is 10 cm.
Cheatsheet
Similar: equal angles and sides in the same ratio
Length scale factor k = new length ÷ corresponding original length
Two pairs of equal angles → similar triangles
A line parallel to a side of a triangle makes a similar triangle
How to answer each type of question
Find missing lengths in similar shapes
2 marks each5
Find a pair of corresponding sides you know both of.
Work out the scale factor (keep it as a fraction if it is not exact).
Multiply to go small to large; divide to go large to small.
Example. Triangle ABC is similar to triangle PQR. AB corresponds to PQ, BC to QR and AC to PR. AB = 6 cm, BC = 9 cm and PQ = 10 cm. (a) Work out the length of QR. (2 marks) (b) PR = 12.5 cm. Work out the length of AC. (2 marks)
Show the model answer
(a) Scale factor = 10 ÷ 6 = \(\frac{5}{3}\), so QR = 9 × \(\frac{5}{3}\) (M1) = 15 cm (A1) (b) AC = 12.5 ÷ \(\frac{5}{3}\) (M1) = 7.5 cm (A1)
Similar triangles from parallel lines
3 marks5
Redraw the small triangle and the large triangle separately, the same way round.
Use whole side lengths for the large triangle.
Find the scale factor and use it.
Example. In triangle ABC, D is a point on AB and E is a point on AC so that DE is parallel to BC. AD = 4 cm, DB = 6 cm and DE = 5 cm. Work out the length of BC.
Show the model answer
AB = 4 + 6 = 10 cm (M1) Scale factor = 10 ÷ 4 = 2.5 (M1) BC = 5 × 2.5 = 12.5 cm (A1)
Shortcuts and memory tricks
Redraw overlapping triangles as two separate triangles, the same way round, and label matching vertices.
Sense check: if the new shape is bigger, your answer must be bigger than the original length.
Keep scale factors as fractions (e.g. 5/3) to avoid rounding errors.
Higher: 'length k, area k², volume k³'. The power matches the units: cm, cm², cm³.
Where marks are lost
Using the short piece DB instead of the whole side AB when a small triangle sits inside a large one.
Adding the difference in lengths instead of multiplying by the scale factor.
Matching the wrong sides when one triangle is turned or flipped.
Higher: using the length scale factor for an area or a volume.
Higher: square rooting a volume factor instead of cube rooting it.
Exam technique
Write the scale factor clearly (e.g. k = 18 ÷ 12 = 1.5); it is usually the first method mark.
To show triangles are similar, show that the angles are equal, giving a reason for each (e.g. corresponding angles are equal).
Check which way you are going: small to large, multiply; large to small, divide.
Higher: the masses of solids made of the same material scale like volumes, so use k³.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy2 marks
In triangle ABC, angle A = 70° and angle B = 75°. In triangle DEF, angle E = 35° and angle F = 70°.
(a) Which condition shows that triangle ABC is similar to triangle DEF? Tick (✓) one box.[1]
AA (two pairs of equal angles)
SAS (two pairs of sides in the same ratio, and the angles between them equal)
SSS (all three pairs of sides in the same ratio)
None of these: the information does not show that the triangles are similar
(b) Complete this statement so that the letters of the two triangles are in corresponding order. Triangle ABC is similar to triangle ..........[1]
Show the answer and mark scheme
(a)Answer: AA (two pairs of equal angles)
B1 for AA (two pairs of equal angles)
(b)Answer: triangle FDE
B1 for FDE
Worked solution: Angle C = 180 − 70 − 75 = 35° and angle D = 180 − 35 − 70 = 75°. So angle A = angle F (70°), angle B = angle D (75°) and angle C = angle E (35°): triangle ABC is similar to triangle FDE.
Question 2Medium2 marks
In triangle ABC, AB = 6 cm and AC = 12 cm. In triangle DEF, DE = 4 cm and DF = 9 cm. Angle BAC = angle EDF = 60°.
(a) Which condition shows that triangle ABC is similar to triangle DEF? Tick (✓) one box.[1]
AA (two pairs of equal angles)
SAS (two pairs of sides in the same ratio, and the angles between them equal)
SSS (all three pairs of sides in the same ratio)
None of these: the information does not show that the triangles are similar
(b) Give a reason why this information does not show that the triangles are similar.[1]
Show the answer and mark scheme
(a)Answer: None of these: the information does not show that the triangles are similar
B1 for None of these: the information does not show that the triangles are similar
(b)Answer: \(\frac{AB}{DE} = \frac{6}{4} = \frac{3}{2}\) but \(\frac{AC}{DF} = \frac{12}{9} = \frac{4}{3}\): the sides either side of the equal angles are not in the same ratio, so SAS does not apply.
C1 for a correct reason with working, e.g. AB ÷ DE = \(\frac{3}{2}\) but AC ÷ DF = \(\frac{4}{3}\), so the sides are not in the same ratio
Worked solution: \(\frac{6}{4} = \frac{3}{2}\) and \(\frac{12}{9} = \frac{4}{3}\). These are different, so the two pairs of sides next to the equal angles are not in the same ratio and the triangles are not similar by SAS.