Area and Volume of Similar Figures and Solids practice questions

Secondary 3 topics from the O-Level Mathematics 4052 syllabus. Every question below has a full worked solution.

When a student gets a question wrong in Math Amigo, it does not show the answer straight away. It reads the working, identifies the step that went wrong, and asks a question that points at the correction. The worked solution is there once a genuine attempt has been made.

Practise area and volume of similar figures and solids free

What this topic covers

Worked examples

Example 1

Two similar triangles have perimeters 1010 cm and 3030 cm. The area of the larger figure is 7272 cm². Find the area of the smaller figure.
Show the worked solution
  • Perimeter ratio $= \frac{P_1}{P_2} = \frac{10}{30} = \frac{2}{6} \quad \scriptsize\textit{perimeter ratio = length ratio}$
  • $\frac{A_1}{A_2} = \left(\frac{P_1}{P_2}\right)^2 \quad \scriptsize\textit{area ratio = square of length ratio}$
  • $\frac{A_1}{72} = \left(\frac{2}{6}\right)^2$
  • $\frac{A_1}{72} = \frac{4}{36}$
  • $A_1 = 72 \times \frac{4}{36}$
  • $= 8$ cm²

Answer: 88

Example 2

Two similar spheres have diameters 66 cm and 99 cm respectively. The volume of the smaller sphere is 6464 cm³. Find the volume of the larger sphere.
Show the worked solution
  • $\text{For similar spheres, the ratio of volumes equals the cube of the ratio of diameters.}$
  • $\frac{V_2}{V_1} = \left(\frac{d_2}{d_1}\right)^3 \quad \scriptsize\textit{volume ratio of similar solids}$
  • $\frac{V_2}{64} = \left(\frac{9}{6}\right)^3$
  • $\frac{V_2}{64} = \left(\frac{3}{2}\right)^3$
  • $\frac{V_2}{64} = \frac{27}{8}$
  • $V_2 = 64 \times \frac{27}{8}$
  • $V_2 = 216$ cm³

Answer: 216216

Example 3

Two similar water jugs have volumes in the ratio 125:8125:8. Find the ratio of their surface areas.
Show the worked solution
  • $\text{Volume ratio} = 125:8 \quad \scriptsize\textit{given}$
  • $\text{Length ratio} = \sqrt[3]{\frac{125}{8}} \quad \scriptsize\textit{length ratio is cube root of volume ratio}$
  • $= \frac{5}{2}$
  • $\text{Surface area ratio} = \left(\frac{5}{2}\right)^2 \quad \scriptsize\textit{area ratio is square of length ratio}$
  • $= \frac{25}{4}$
  • $= 25:4$

Answer: 25:425:4

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Which syllabus does this follow?

O-Level Mathematics (4052), for Singapore secondary students. Math Amigo covers Secondary 1 to 4, both O-Level Mathematics and Additional Mathematics.

Will it just show my child the answer?

No. On a wrong answer it looks at the working, points to the step that went wrong, and asks a question that helps the student find the correction. The full worked solution is available after a genuine attempt.

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