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 school in Brisbane, Queensland, is ordering materials for a new classroom display. The display is designed as a composite shape consisting of a rectangle measuring 62 cm by 44 cm, with a right-angled triangle attached to one of its 62 cm sides.

The triangle has legs of 62 cm and 44 cm, and is attached so that the 62 cm leg aligns with the rectangle's side and the right angle is at the corner of the rectangle. The triangle's hypotenuse is not part of the outer boundary.

What is 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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Formula

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 one side. The rectangle has dimensions 62 cm by 44 cm.
  2. The area of a rectangle is given by length × width. Substitute the values: 62 cm × 44 cm.
  3. Calculate the rectangle's area: 62 × 44 = (60 + 2) × 44 = 60×44 + 2×44 = 2640 + 88 = 2728 cm².
  4. The triangle is right-angled with legs of 62 cm and 44 cm. The area of a right-angled triangle is (1/2) × base × height.
  5. Substitute the base and height: (1/2) × 62 cm × 44 cm.
  6. Calculate: (1/2) × 62 × 44 = 31 × 44. Compute 31 × 44 = (30 + 1) × 44 = 30×44 + 1×44 = 1320 + 44 = 1364 cm².
  7. The total area of the composite shape is the sum of the rectangle and triangle areas: 2728 cm² + 1364 cm².
  8. Add: 2728 + 1364 = 4092 cm².
  9. The shape is composed of two parts: a rectangle and a right triangle sharing a side. No overlapping or missing regions are present.
  10. The triangle is attached along the 62 cm side, and its legs are 62 cm and 44 cm, so the area calculation is valid.
  11. The total area is 4092 cm². This is an exact integer value, so no simplification or radical form is needed.
  12. The exact area of the composite shape is 4092 square centimetres.

Reference answer: 4092 cm²

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