NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Area and perimeter of composite shapes

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Required skills (click to reveal)

A sports training facility in Brisbane, Queensland, is installing a new composite-shaped mat for agility drills. The mat consists of a central rectangle measuring 22 metres by 43 metres, with a right-angled triangular section attached to one of the 43-metre sides.

The triangle has legs of 43 metres and 22 metres, and is attached so that the 43-metre leg aligns with the rectangle's side and the right angle is at the corner of the rectangle. The hypotenuse of the triangle 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. The rectangle has length 43 m and width 22 m. The triangle has legs of 43 m and 22 m, attached along the 43 m side of the rectangle.
  2. The area of a rectangle is given by length × width. Substitute the values: 43 m × 22 m.
  3. Calculate the rectangle area: 43 × 22 = 946. So, area = 946 square metres.
  4. The area of a right-angled triangle is (1/2) × base × height. The base and height are 43 m and 22 m, respectively.
  5. Substitute values: (1/2) × 43 × 22.
  6. First, compute 43 × 22 = 946. Then, (1/2) × 946 = 473. So, triangle area = 473 square metres.
  7. The total area of the composite shape is the sum of the rectangle and triangle areas: 946 + 473.
  8. Add: 946 + 473 = 1419. So, total area = 1419 square metres.
  9. The problem asks for the exact area. 1419 is an integer, so it is already exact.
  10. Verify the shape: the triangle is attached along the 43 m side, and its legs are 43 m and 22 m, so the right angle is at the corner. This matches the description.
  11. Check calculation: 43 × 22 = 946 (correct), (1/2) × 946 = 473 (correct), 946 + 473 = 1419 (correct).
  12. The total area of the composite shape is 1419 square metres.

Reference answer: 1419 square metres

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