NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Two-dimensional representations

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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 base and a square pyramid roof. The base of the pyramid is attached to the top face of the rectangular prism, which measures 8 m in length, 6 m in width, and 3 m in height.

The pyramid has a square base matching the top face of the prism (8 m by 8 m) and a vertical height of 5 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

  1. The top view of the structure is a two-dimensional representation showing the footprint of the base and the projection of the pyramid's base. The base is a rectangular prism with dimensions 8 m (length) by 6 m (width).
  2. The pyramid has a square base that matches the top face of the prism. However, the prism's top face is 8 m by 6 m, which is not square. The pyramid's base is specified as 8 m by 8 m, so it extends beyond the prism's top face in the width direction.
  3. The top view shows the full footprint of the rectangular prism base (8 m × 6 m) and the full projection of the pyramid's base (8 m × 8 m). Since the pyramid is placed on top of the prism, its base extends beyond the prism's top face in the width direction.
  4. The total area of the top view is the area of the rectangular prism base plus the area of the pyramid base, minus the overlapping region. However, since the pyramid base extends beyond the prism base, the total footprint is not simply additive.
  5. The pyramid base is 8 m × 8 m and is placed so that its base aligns with the top face of the prism. The prism's top face is 8 m × 6 m, so the pyramid base extends 1 m beyond the prism in the width direction (8 m - 6 m = 2 m, so 1 m on each side).
  6. The top view shows the full 8 m × 8 m square of the pyramid base, because the pyramid is built on top and its base is visible from above. The prism base is entirely within the pyramid base's footprint.
  7. Therefore, the two-dimensional top view is simply the area of the pyramid's square base, since it fully covers the prism base and extends beyond it.
  8. The area of a square is given by side × side. The side length of the pyramid base is 8 m.
  9. Substitute: area = 8 m × 8 m = 64 m².
  10. The area of the top view is 64 square metres.
  11. Check: the pyramid base is 8 m × 8 m, and since it is visible from above and covers the prism base, the total footprint is 64 m².
  12. The top view shows the full 8 m × 8 m square of the pyramid base, so the area is 64 m².

Reference answer: 64

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