NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Surface area and volume

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A right triangular prism used in a sustainable architecture project in Canberra has a triangular base with side lengths of 9 m, 12 m, and 15 m, and a height of 10 m. The base is a right-angled triangle with the right angle between the 9 m and 12 m sides.

Calculate the exact surface area of the prism in square metres. Use the formula SA = 2 × (area of base) + (perimeter of base) × (height of prism).

Provide the exact answer as a simplified fraction or integer.

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Explanation

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Worked solution

  1. The prism has a right-angled triangular base with sides 9 m, 12 m, and 15 m. The right angle is between the 9 m and 12 m sides, so these are the legs of the triangle.
  2. The area of a right-angled triangle is (1/2) × base × height. Use base = 9 m and height = 12 m.
  3. Area of base = (1/2) × 9 × 12 = (1/2) × 108 = 54 m².
  4. The perimeter of the base is the sum of all three sides: 9 m + 12 m + 15 m = 36 m.
  5. The height of the prism is 10 m, which is the distance between the two triangular bases.
  6. Use the surface area formula: SA = 2 × (area of base) + (perimeter of base) × (height of prism).
  7. Substitute values: SA = 2 × 54 + 36 × 10.
  8. Calculate 2 × 54 = 108.
  9. Calculate 36 × 10 = 360.
  10. Add: 108 + 360 = 468.
  11. The surface area is 468 m². This includes both triangular bases and the three rectangular lateral faces.
  12. The final answer is 468 square metres, which is an integer and already simplified.

Reference answer: 468

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