Similar lengths, areas and volumes Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Similar lengths, areas and volumes questions. See exactly how to solve problems on area scale factor, similar shapes, volume scale factor, similar solids.

area scale factorsimilar shapesvolume scale factorsimilar solidsratio of areas or volumesscale models
GCSE Higher70 questionsStep-by-step solutions
Question 1
2 markseasy
Two mathematically similar shapes have corresponding lengths in the ratio 1:31 : 3. The area of the smaller shape is 55 cm2^2. Work out the area of the larger shape.

Worked solution

  1. Work out the length scale factor.

    k=31=3k = \frac{3}{1} = 3

    The corresponding lengths are in the ratio 1:31 : 3. Going from the smaller shape to the larger shape, every length is multiplied by 31=3\frac{3}{1} = 3, and that is the LENGTH scale factor kk.

  2. Work out the area scale factor.

    k2=(3)2=9k^2 = \left(3\right)^2 = 9

    Areas scale by k2k^2, so the area scale factor is (3)2=9\left(3\right)^2 = 9.

  3. State the area of the larger shape.

    A=5×9=45 cm2A = 5 \times 9 = 45\text{ cm}^2

    The area of the larger shape is 4545 square centimetres.

Answer
A=45 cm2A = 45\text{ cm}^2
Question 2
2 markseasy
Two mathematically similar shapes have corresponding lengths in the ratio 2:32 : 3. The area of the smaller shape is 2020 cm2^2. Work out the area of the larger shape.

Worked solution

  1. Work out the length scale factor.

    k=32=1.5k = \frac{3}{2} = 1.5

    The corresponding lengths are in the ratio 2:32 : 3. Going from the smaller shape to the larger shape, every length is multiplied by 32=1.5\frac{3}{2} = 1.5, and that is the LENGTH scale factor kk.

  2. Work out the area scale factor.

    k2=(1.5)2=2.25k^2 = \left(1.5\right)^2 = 2.25

    Areas scale by k2k^2, so the area scale factor is (1.5)2=2.25\left(1.5\right)^2 = 2.25.

  3. State the area of the larger shape.

    A=20×2.25=45 cm2A = 20 \times 2.25 = 45\text{ cm}^2

    The area of the larger shape is 4545 square centimetres.

Answer
A=45 cm2A = 45\text{ cm}^2
Question 3
2 markseasy
Two mathematically similar shapes have corresponding lengths in the ratio 1:41 : 4. The area of the smaller shape is 33 cm2^2. Work out the area of the larger shape.

Worked solution

  1. Work out the length scale factor.

    k=41=4k = \frac{4}{1} = 4

    The corresponding lengths are in the ratio 1:41 : 4. Going from the smaller shape to the larger shape, every length is multiplied by 41=4\frac{4}{1} = 4, and that is the LENGTH scale factor kk.

  2. Work out the area scale factor.

    k2=(4)2=16k^2 = \left(4\right)^2 = 16

    Areas scale by k2k^2, so the area scale factor is (4)2=16\left(4\right)^2 = 16.

  3. State the area of the larger shape.

    A=3×16=48 cm2A = 3 \times 16 = 48\text{ cm}^2

    The area of the larger shape is 4848 square centimetres.

Answer
A=48 cm2A = 48\text{ cm}^2
Question 4
2 markseasy
Two mathematically similar shapes have corresponding lengths in the ratio 1:31 : 3. The area of the larger shape is 6363 cm2^2. Work out the area of the smaller shape.

Worked solution

  1. Work out the length scale factor.

    k=13=13k = \frac{1}{3} = \frac{1}{3}

    The corresponding lengths are in the ratio 1:31 : 3. Going from the larger shape to the smaller shape, every length is multiplied by 13=13\frac{1}{3} = \frac{1}{3}, and that is the LENGTH scale factor kk.

  2. Work out the area scale factor.

    k2=(13)2=19k^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9}

    Areas scale by k2k^2, so the area scale factor is (13)2=19\left(\frac{1}{3}\right)^2 = \frac{1}{9}.

  3. State the area of the smaller shape.

    A=63×19=7 cm2A = 63 \times \frac{1}{9} = 7\text{ cm}^2

    The area of the smaller shape is 77 square centimetres.

Answer
A=7 cm2A = 7\text{ cm}^2
Question 5
2 markseasy
Two mathematically similar shapes have corresponding lengths in the ratio 2:52 : 5. The area of the larger shape is 100100 cm2^2. Work out the area of the smaller shape.

Worked solution

  1. Work out the length scale factor.

    k=25=0.4k = \frac{2}{5} = 0.4

    The corresponding lengths are in the ratio 2:52 : 5. Going from the larger shape to the smaller shape, every length is multiplied by 25=0.4\frac{2}{5} = 0.4, and that is the LENGTH scale factor kk.

  2. Work out the area scale factor.

    k2=(0.4)2=0.16k^2 = \left(0.4\right)^2 = 0.16

    Areas scale by k2k^2, so the area scale factor is (0.4)2=0.16\left(0.4\right)^2 = 0.16.

  3. State the area of the smaller shape.

    A=100×0.16=16 cm2A = 100 \times 0.16 = 16\text{ cm}^2

    The area of the smaller shape is 1616 square centimetres.

Answer
A=16 cm2A = 16\text{ cm}^2

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