Robinson Tuition
Grade 9 · Higher · free GCSE maths worksheet

Problems on similar shapes with algebra

Area and volume ratios leading to equations.

8 original questions, 32 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Two shapes are mathematically similar. The smaller has area 12 cm2 and the larger has area 27 cm2. A side of the smaller shape is 4 cm long.

Work out the length of the matching side of the larger shape.

2 marks
Show answer
6 cm
2

Triangle P and triangle Q are mathematically similar. Triangle P has area 20 cm2 and base x cm. Triangle Q has area 45 cm2 and the matching base is (x + 3) cm.

Work out the value of x.

4 marks
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x = 6
3

Two solid shapes are mathematically similar. The smaller has volume 54 cm3 and height 6 cm. The larger has volume 250 cm3 and height (2x + 4) cm.

Work out the value of x.

4 marks
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x = 3
4

In triangle ABC, D lies on AB and E lies on AC so that DE is parallel to BC.

AD : DB = 2 : 1.

The area of triangle ADE is 24 cm2.

Work out the area of the trapezium DBCE.

ABCDEDiagram NOT accurately drawn
4 marks
Show answer
30 cm²
5

Two rectangles are mathematically similar. The smaller has area 36 cm2 and the larger has area 100 cm2. The length of the smaller is (x + 1) cm and the matching length of the larger is 2x cm.

Work out the value of x and the two lengths.

4 marks
Show answer
x) 5 lengths) 6 cm and 10 cm
6

In triangle ABC, the point D is on AB and the point E is on AC so that DE is parallel to BC. AD = x cm and AB = 3x cm.

The area of triangle ADE is A cm2.

Show that the area of the trapezium DBCE is 8A cm2.

4 marks
Show answer
ABC is an enlargement of ADE with scale factor 3, so area ABC = 9A and DBCE = 9A - A = 8A
7

Two mathematically similar containers, A and B, have volumes 400 cm3 and 1350 cm3.

  1. Container A has height 12 cm. Work out the height of container B.
  2. Container B has surface area 468 cm2. Work out the surface area of container A.
5 marks
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a) 18 cm b) 208 cm²
8

Two solid shapes, S and T, are mathematically similar. Their volumes are in the ratio 8 : 27.

The height of S is x cm and the height of T is (x + 4) cm. The surface area of S is 96 cm2.

  1. Work out the value of x.
  2. Work out the surface area of T.
5 marks
Show answer
a) 8 b) 216 cm²

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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