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

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A walking route in Melbourne, Victoria, is designed as a composite shape consisting of a rectangular path measuring 48 metres in length and 12 metres in width, with a right-angled triangular extension attached to one of its 48-metre sides. The triangle has legs of 48 metres and 57 metres, and is attached so that the 48-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. What is the exact total area of the walking route in square metres?

Express your answer as an exact fraction or simplified radical expression.

Concept illustrations

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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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Worked solution

  1. The walking route consists of a rectangle and a right-angled triangle attached to one of its 48-metre sides. The rectangle has length 48 m and width 12 m. The triangle has legs of 48 m and 57 m, with the 48-metre leg aligned to the rectangle's side and the right angle at the corner.
  2. The total area is the sum of the area of the rectangle and the area of the triangle.
  3. Area of rectangle = length × width = 48 m × 12 m.
  4. 48 × 12 = 576. So, area of rectangle = 576 square metres.
  5. Area of right-angled triangle = (1/2) × base × height. The base and height are the two legs: 48 m and 57 m.
  6. Area = (1/2) × 48 × 57.
  7. First, compute 48 × 57. 48 × 50 = 2400, 48 × 7 = 336. So 2400 + 336 = 2736.
  8. Now divide by 2: 2736 ÷ 2 = 1368. So, area of triangle = 1368 square metres.
  9. Total area = area of rectangle + area of triangle = 576 + 1368.
  10. 576 + 1368 = 1944. So, total area = 1944 square metres.
  11. The total area is 1944 m². This is an integer, so it is already in exact form as a whole number.
  12. The exact total area of the walking route is 1944 square metres.

Reference answer: 1944 m²

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