Year 9 Surface Area & Volume Worksheets

Worked examples

Year 9 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 cylinder

   A cylinder has a radius of $4$ cm and a height of $10$ cm. Find its total surface area. Give your answer correct to one decimal place.

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   Key idea: The total surface area of a cylinder is the curved surface (a rectangle when unrolled) plus two circular ends: $\text{TSA} = 2\pi r h + 2\pi r^2$.

   1. Write the formula: $\text{TSA} = 2\pi r h + 2\pi r^2$
   2. Substitute $r = 4$ and $h = 10$:
      $\text{TSA} = 2\pi(4)(10) + 2\pi(4)^2$
   3. Simplify each term:
      $\text{TSA} = 80\pi + 32\pi = 112\pi$
   4. Evaluate:
      $\text{TSA} = 112\pi \approx 351.9 \text{ cm}^2$

   Answer: $351.9$ cm²

2. Example 2 · Volume of a composite solid (prism and cylinder)

   A composite solid is made by placing a cylinder on top of a rectangular prism. The rectangular prism has a length of $12$ cm, a width of $12$ cm and a height of $8$ cm. The cylinder sits centred on top and has a radius of $3$ cm and a height of $5$ cm. Find the total volume of the solid. Give your answer correct to one decimal place.

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   Key idea: The total volume of a composite solid is the sum of the volumes of each part: $V_{\text{total}} = V_{\text{prism}} + V_{\text{cylinder}}$.

   1. Find the volume of the rectangular prism:
      $V_{\text{prism}} = l \times w \times h = 12 \times 12 \times 8 = 1152 \text{ cm}^3$
   2. Find the volume of the cylinder:
      $V_{\text{cylinder}} = \pi r^2 h = \pi \times 3^2 \times 5 = 45\pi \text{ cm}^3$
   3. Add the two volumes:
      $V_{\text{total}} = 1152 + 45\pi \approx 1152 + 141.4 = 1293.4 \text{ cm}^3$

   Answer: $1293.4$ cm³

3. Example 3 · Real-world problem: painting a room

   Mia wants to paint the four walls and the ceiling of a rectangular room. The room is $6$ m long, $4$ m wide and $2.5$ m high. The room has one door measuring $2$ m by $0.9$ m and one window measuring $1.2$ m by $1$ m, which are not painted. One litre of paint covers $12$ m² and costs $8.50. How much will Mia spend on paint? Round up to the nearest whole litre when buying paint.

   Show worked solution

   Key idea: Find the total area to be painted by calculating the area of all four walls and the ceiling, then subtract the unpainted door and window. Use the coverage rate to find litres needed, then calculate the cost.

   1. Find the area of the four walls:
      $A_{\text{walls}} = 2(lh + wh) = 2(6 \times 2.5 + 4 \times 2.5) = 2(15 + 10) = 50 \text{ m}^2$
   2. Find the area of the ceiling:
      $A_{\text{ceiling}} = 6 \times 4 = 24 \text{ m}^2$
   3. Subtract the door and window:
      $A_{\text{painted}} = 50 + 24 - (2 \times 0.9) - (1.2 \times 1) = 74 - 1.8 - 1.2 = 71 \text{ m}^2$
   4. Find litres needed, rounding up:
      $\text{Litres} = \lceil 71 \div 12 \rceil = \lceil 5.916\ldots \rceil = 6 \text{ L}$
   5. Calculate the cost:
      $\text{Cost} = 6 \times \$8.50 = \$51.00$

   Answer: $51.00

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