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A three-dimensional model of a building in Sydney is composed of a rectangular prism base measuring 12 m in length, 8 m in width, and 5 m in height, with a right triangular prism roof attached to the longer side. The triangular roof has a base equal to the length of the building (12 m) and a height of 4 m.
When viewed from the front (looking at the 12 m by 5 m face), the two-dimensional representation shows a composite shape consisting of a rectangle and a triangle on top.
What is the exact area, in square metres, of this two-dimensional front view?
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Worked solution
- The front view of the building consists of a rectangle and a triangle stacked vertically. The rectangle represents the front face of the base, which is 12 m long and 5 m high.
- The area of the rectangular part is calculated using the formula: area = length × height. Substituting the values: area = 12 m × 5 m.
- Performing the multiplication: 12 × 5 = 60. The area of the rectangle is 60 square metres.
- The triangular roof has a base of 12 m (same as the building length) and a height of 4 m. This triangle sits on top of the rectangle.
- The area of a triangle is given by: area = (1/2) × base × height. Substituting: area = (1/2) × 12 m × 4 m.
- First, compute (1/2) × 12 = 6. Then, 6 × 4 = 24. The area of the triangle is 24 square metres.
- The total area of the front view is the sum of the rectangular base and the triangular roof: 60 m² + 24 m².
- Adding: 60 + 24 = 84. The total area is 84 square metres.
- The front view is a composite shape made of a rectangle and a triangle. No overlapping or missing parts are present.
- The representation is two-dimensional and matches the given 3D structure when viewed from the front.
- The calculation uses only given dimensions and standard area formulas. No approximations were made.
- The total area of the front view is 84 square metres.
Reference answer: 84
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