NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Area and perimeter of composite shapes

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Required skills (click to reveal)

A composite shape in Darwin, Northern Territory, is formed by attaching a rectangle measuring 14 metres by 32 metres to a right-angled triangle with legs of 14 metres and 32 metres, such that the triangle is attached along the 32-metre side of the rectangle and the right angle is at the corner of the rectangle. The triangle's hypotenuse is not part of the outer boundary.

Calculate the exact area of the composite shape in square metres. Express your answer as an exact fraction or simplified radical expression.

Concept illustrations

These are separate worked examples of related concepts, not a diagram of this question. Their values and dimensions do not describe your question.

Subtract a rectangular corner cut-out. The outer rectangle is 10 cm by 6 cm. Removing a 2 cm by 3 cm top-right corner leaves 60 − 6 = 54 cm². All dimensions are supplied; drawing is to one consistent scale.10 cm6 cm2 cm3 cm54 cm²
The outer rectangle is 10 cm by 6 cm. Removing a 2 cm by 3 cm top-right corner leaves 60 − 6 = 54 cm². All dimensions are supplied; drawing is to one consistent scale.
Area and perimeter of a rectangle. 6 × 4 = 24 square units inside; 6 + 4 + 6 + 4 = 20 units around.6 units across4
6 × 4 = 24 square units inside; 6 + 4 + 6 + 4 = 20 units around.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The composite shape consists of a rectangle and a right-angled triangle attached along the 32-metre side. The rectangle has dimensions 14 m by 32 m.
  2. The area of a rectangle is given by length × width. Here, length = 32 m and width = 14 m.
  3. Substitute values: Area_rectangle = 32 × 14. Calculate: 32 × 14 = 448 square metres.
  4. The right-angled triangle has legs of 14 m and 32 m, and is attached along the 32-metre side of the rectangle. The right angle is at the corner of the rectangle.
  5. The area of a right-angled triangle is (1/2) × base × height. Here, base = 32 m and height = 14 m.
  6. Substitute values: Area_triangle = (1/2) × 32 × 14. First, (1/2) × 32 = 16. Then, 16 × 14 = 224 square metres.
  7. The total area of the composite shape is the sum of the rectangle and triangle areas: 448 + 224.
  8. Add: 448 + 224 = 672 square metres.
  9. The triangle's hypotenuse is not part of the outer boundary, so no adjustment is needed to the area calculation.
  10. The area is exact and does not require a radical or fractional form since 672 is an integer.
  11. Check: 32 × 14 = 448 (correct), (1/2) × 32 × 14 = 224 (correct), 448 + 224 = 672 (correct).
  12. The total area of the composite shape is 672 square metres.

Reference answer: 672 square metres

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