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 composite shape in Brisbane, Queensland, is formed by attaching a rectangle measuring 88 cm by 95 cm to a right-angled triangle with legs of 88 cm and 95 cm, such that the triangle is attached along the 95 cm 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 centimetres. 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 95 cm side. The rectangle has dimensions 88 cm by 95 cm.
  2. The area of a rectangle is given by length × width. Substitute the values: 88 cm × 95 cm.
  3. Calculate 88 × 95: 88 × (100 - 5) = 88 × 100 - 88 × 5 = 8800 - 440 = 8360 cm².
  4. The right-angled triangle has legs of 88 cm and 95 cm, and is attached along the 95 cm side. The area of a right-angled triangle is (1/2) × base × height.
  5. Substitute base = 88 cm, height = 95 cm: (1/2) × 88 × 95.
  6. Calculate (1/2) × 88 × 95: first, 88 × 95 = 8360, then divide by 2: 8360 / 2 = 4180 cm².
  7. The total area of the composite shape is the sum of the rectangle and triangle areas: 8360 cm² + 4180 cm².
  8. Add: 8360 + 4180 = 12,540 cm².
  9. The problem asks for the exact area. 12,540 is an integer, so it is already exact.
  10. Verify the shape: the triangle is attached along the 95 cm side, and the right angle is at the corner, so no overlapping or missing parts.
  11. The hypotenuse is not part of the outer boundary, but this does not affect area calculation since area is internal.
  12. The total area of the composite shape is 12,540 square centimetres.

Reference answer: 12540 cm²

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