Similar Shapes (Area and Volume)
Similar shapes (area and volume) covers how the linear scale factor between two mathematically similar shapes links to their area and volume scale factors: if lengths scale by k, areas scale by k^2 and volumes scale by k^3. It lets you find missing lengths, areas or volumes, including in reverse using square or cube roots. This is a Higher GCSE geometry topic mixing 2D and 3D shapes.
Before you start
Make sure you're comfortable with these topics first:
Method
- Confirm the shapes are mathematically similar (corresponding angles equal, corresponding sides in the same ratio) before using any scale factor shortcuts.
- Find the linear (length) scale factor by dividing a length on the new shape by the corresponding length on the original shape.
- To scale an area, square the linear scale factor; to scale a volume, cube the linear scale factor.
- Multiply the original area or volume by the appropriate scale factor to get the new one.
- To work backwards from an area ratio to a linear scale factor, take the square root; from a volume ratio, take the cube root.
- Always check which scale factor the question has actually given you (linear, area or volume) before using it.
Worked example
Two mathematically similar containers, A and B, have a linear scale factor of 3 from A to B. The volume of container A is 20 cm^3. Work out the volume of container B.
- The linear scale factor from A to B is 3.
- To scale a volume, cube the linear scale factor: 3^3 = 27.
- Multiply the volume of A by this volume scale factor: 20 x 27 = 540.
- The volume of container B is 540 cm^3.
Practice questions
Try each question, then tap to reveal the answer.
Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
Cylinder X and cylinder Y are mathematically similar. Cylinder X has height 6 cm and cylinder Y has height 15 cm. The surface area of cylinder X is 40 cm^2. Work out the surface area of cylinder Y.
Shape G and shape H are mathematically similar. The area of shape G is 24 cm^2 and the area of shape H is 54 cm^2. A side of shape G is 8 cm. (a) Work out the linear scale factor from shape G to shape H. (b) Work out the length of the corresponding side on shape H.
A scale model of a statue is built using a scale of 1:20 (every linear dimension of the real statue is 20 times the corresponding dimension of the model). The model has a volume of 45 cm^3 and a surface area of 130 cm^2. (a) Work out the volume of the real statue, giving your answer in cm^3 in standard form correct to 3 significant figures. (b) Work out the surface area of the real statue in m^2, correct to 3 significant figures.
Free printable worksheet
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