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 structure in Brisbane is composed of a square pyramid on top of a cube. The cube has a side length of 6 m, and the pyramid has a square base matching the top face of the cube and a vertical height of 4 m.

When viewed from the front, the two-dimensional representation shows a composite shape consisting of a square and an isosceles triangle on top.

What is the exact area, in square metres, of this two-dimensional front view?

Concept illustrations

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A cube net with six square faces. Four squares in a row, with one above and one below the second square, form a valid cube net. Each face has the same side length.1234566 faces
Four squares in a row, with one above and one below the second square, form a valid cube net. Each face has the same side length.
A rectangular prism. A prism of length 4 cm, width 3 cm and height 2 cm has volume 24 cm³ = 24 mL. The perspective drawing is schematic.4 cm3 cm2 cm
A prism of length 4 cm, width 3 cm and height 2 cm has volume 24 cm³ = 24 mL. The perspective drawing is schematic.

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The front view of the 3D structure consists of a square (from the cube) and an isosceles triangle (from the pyramid) stacked vertically. The base of the triangle matches the top face of the square.
  2. The side length of the cube is 6 m, so the square in the front view has a width and height of 6 m.
  3. The pyramid has a vertical height of 4 m, so the height of the isosceles triangle in the front view is 4 m.
  4. The base of the triangle is equal to the side of the square, which is 6 m.
  5. The area of the square is calculated using the formula: area = side × side. Substituting: 6 m × 6 m = 36 m².
  6. The area of the triangle is calculated using the formula: area = (1/2) × base × height. Substituting: (1/2) × 6 m × 4 m = 12 m².
  7. The total area of the front view is the sum of the square and triangle areas: 36 m² + 12 m² = 48 m².
  8. The front view is a composite shape with no overlapping regions, so the areas add directly.
  9. The result is exact because all given values are integers and the area formulas are exact.
  10. The front view shows the full height of the cube (6 m) and the full height of the pyramid (4 m), so the total height is 10 m, but this is not needed for area calculation.
  11. The area of the front view is 48 square metres, as it is composed of a 6 m × 6 m square and a 6 m base × 4 m height triangle.
  12. Final verification: 36 m² (square) + 12 m² (triangle) = 48 m². No units are missing or incorrect.

Reference answer: 48

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