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A three-dimensional model of a traditional Aboriginal cultural map from the Northern Territory is represented using a combination of a rectangular prism and a square pyramid. The base of the pyramid is attached to the top face of the rectangular prism, which measures 10 m in length, 6 m in width, and 4 m in height.
The pyramid has a square base matching the top face of the prism (10 m by 10 m) and a vertical height of 6 m. When viewed from directly above, the two-dimensional representation of the entire structure shows the footprint of the base and the projection of the pyramid's base.
What is the exact area, in square metres, of this two-dimensional top view?
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Worked solution
- The structure consists of a rectangular prism and a square pyramid stacked vertically, with the pyramid's base attached to the top face of the prism.
- The top view is a two-dimensional representation of the footprint of the entire structure as seen from directly above.
- The base of the rectangular prism is 10 m long and 6 m wide, so its footprint is a rectangle of 10 m × 6 m.
- The square pyramid has a base that matches the top face of the prism, which is 10 m × 10 m. However, since the prism's top face is only 10 m × 6 m, the pyramid's base extends beyond the prism's footprint in the width direction.
- The footprint of the pyramid is a square of 10 m × 10 m, but it is positioned so that its base covers the 10 m × 6 m top face of the prism and extends 2 m beyond on each side in the width direction.
- The top view shows the combined footprint of the prism and the pyramid. Since the pyramid's base is larger than the prism's top face, the total footprint is the area of the pyramid's base, because it completely covers the prism's footprint and extends beyond.
- The pyramid's base is a square with side length 10 m, so its area is 10 m × 10 m.
- Calculate the area: 10 × 10 = 100 square metres.
- The prism's footprint (10 m × 6 m = 60 m²) is entirely covered by the pyramid's base, so it does not contribute additional area to the top view.
- Therefore, the total area of the two-dimensional top view is the area of the pyramid's base, which is 100 m².
- Check: 10 m × 10 m = 100 m². The pyramid's base fully covers the prism's top face and extends beyond, so the visible footprint is the full 10 m × 10 m square.
- The top view shows a 10 m × 10 m square, so the area is exactly 100 square metres.
Reference answer: 100
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