Year 10 Surface Area & Volume Worksheets

Worked examples

Year 10 surface area and volume: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

1. Example 1 · Surface area of a composite solid (prism on top of prism)

   A solid is made by placing a triangular prism on top of a rectangular prism. The rectangular prism is 10 cm long, 6 cm wide and 4 cm tall. The triangular prism sits exactly on the top face of the rectangular prism and runs the full 10 cm length. Its cross-section is an isosceles triangle with base 6 cm and perpendicular height 3 cm; each sloping face has a slant height of 3.35 cm. Find the total surface area of the composite solid.

   Show worked solution

   Key idea: Add the surface areas of each part, then subtract any faces that are joined (hidden) and therefore no longer on the outside.

   1. Rectangular prism faces. The two ends ($6 \times 4$), two long sides ($10 \times 4$), and the bottom ($10 \times 6$). The top face is hidden where the triangular prism sits, so omit it.
      $\text{SA}_{\text{rect}} = 2(6 \times 4) + 2(10 \times 4) + (10 \times 6) = 48 + 80 + 60 = 188 \text{ cm}^2$
   2. Triangular prism faces. Two triangular ends and two sloping rectangles. The base of the triangular prism is hidden (joined to the rectangular prism), so omit it.
      $\begin{gathered}\text{Area of each triangle} = \tfrac{1}{2} \times 6 \times 3 = 9 \text{ cm}^2 \\ \text{Two sloping rectangles} = 2 \times (3.35 \times 10) = 67 \text{ cm}^2 \\ \text{SA}_{\text{tri}} = 2(9) + 67 = 18 + 67 = 85 \text{ cm}^2\end{gathered}$
   3. Total surface area:
      $\text{SA} = 188 + 85 = 273 \text{ cm}^2$ Answer: $273 \text{ cm}^2$

2. Example 2 · Volume of a composite solid (cylinder on top of rectangular prism)

   A solid ornament is made from a rectangular prism with a half-cylinder on top. The rectangular prism is 14 cm long, 8 cm wide and 5 cm tall. The half-cylinder runs along the full 14 cm length and has a diameter of 8 cm. Find the total volume of the ornament. Give your answer correct to one decimal place.

   Show worked solution

   Key idea: The total volume equals the volume of the rectangular prism plus the volume of the half-cylinder.

   1. Volume of the rectangular prism:
      $V_{\text{rect}} = 14 \times 8 \times 5 = 560 \text{ cm}^3$
   2. Radius of the half-cylinder: $r = 8 \div 2 = 4$ cm.
   3. Volume of the half-cylinder:
      $V_{\text{half-cyl}} = \tfrac{1}{2} \times \pi r^2 \times h = \tfrac{1}{2} \times \pi \times 4^2 \times 14 = \tfrac{1}{2} \times 16\pi \times 14 = 112\pi \approx 351.9 \text{ cm}^3$
   4. Total volume:
      $V = 560 + 112\pi \approx 560 + 351.9 = 911.9 \text{ cm}^3$ Answer: $911.9 \text{ cm}^3$

3. Example 3 · Capacity of a water tank with a cylindrical pipe drilled through it

   A rectangular water tank is 1.2 m long, 0.8 m wide and 0.6 m tall. A cylindrical pipe with a diameter of 0.2 m is drilled horizontally through the full 1.2 m length of the tank, removing material from the solid. Find the volume of water the tank can hold, in litres. Give your answer correct to the nearest litre. (Use $1 \text{ m}^3 = 1000 \text{ L}$.)

   Show worked solution

   Key idea: Use the subtraction method. The usable volume equals the volume of the rectangular tank minus the volume removed by the cylindrical pipe.

   1. Volume of the rectangular tank:
      $V_{\text{rect}} = 1.2 \times 0.8 \times 0.6 = 0.576 \text{ m}^3$
   2. Radius of the cylinder: $r = 0.2 \div 2 = 0.1$ m.
   3. Volume of the cylindrical pipe:
      $V_{\text{cyl}} = \pi r^2 h = \pi \times 0.1^2 \times 1.2 = 0.012\pi \approx 0.03770 \text{ m}^3$
   4. Volume of water the tank holds:
      $V = 0.576 - 0.012\pi \approx 0.576 - 0.03770 = 0.53830 \text{ m}^3$
   5. Convert to litres:
      $0.53830 \times 1000 \approx 538 \text{ L}$ Answer: $538 \text{ L}$

Common mistakes

Where students usually lose marks

Try one yourself

Your turn

Got a different surface area and volume question wrong? Upload it to Tuterly and get the worked solution and a fresh one to practise, free.