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Measurement - Calculating perimeter and area of compound shapes

Grade 5IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A compound (or composite) shape is a figure made up of two or more basic geometric shapes, such as rectangles and squares.

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To find missing side lengths, use the lengths of parallel sides (e.g., the total horizontal length equals the sum of the smaller horizontal segments).

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Perimeter is the total distance around the outside edge of the shape; internal lines used for splitting the shape are not counted.

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Area can be calculated by 'Decomposition' (splitting the shape into smaller rectangles and adding their areas) or 'Subtraction' (calculating the area of a large bounding rectangle and subtracting the empty space).

📐Formulae

Perimeter=Sum of all exterior side lengths\text{Perimeter} = \text{Sum of all exterior side lengths}

Area of Rectangle=length×width\text{Area of Rectangle} = \text{length} \times \text{width}

Area of Square=side×side\text{Area of Square} = \text{side} \times \text{side}

💡Examples

Problem 1:

An L-shaped room has the following outer dimensions: a total height of 10m and a total base width of 8m. The 'cut-out' corner has a top horizontal length of 5m and a right-side vertical length of 4m. Find the Perimeter and Area.

Solution:

  1. Find missing sides: Horizontal missing side = 8m−5m=3m8m - 5m = 3m. Vertical missing side = 10m−4m=6m10m - 4m = 6m.
  2. Perimeter: 10+8+4+3+6+5=36m10 + 8 + 4 + 3 + 6 + 5 = 36m.
  3. Area: Split into two rectangles. Rectangle A (top) = 5m×6m=30m25m \times 6m = 30m^2. Rectangle B (bottom) = 8m×4m=32m28m \times 4m = 32m^2. Total Area = 30+32=62m230 + 32 = 62m^2.

Explanation:

First, we identify the lengths of the two 'hidden' sides by subtracting the known segments from the total parallel lengths. For the perimeter, we sum all six outer sides. For the area, we divide the L-shape into two simpler rectangles, calculate their individual areas, and add them together.

Problem 2:

A square of side 3cm is cut out from the corner of a larger rectangle measuring 12cm by 7cm. Calculate the area of the remaining shape.

Solution:

  1. Area of large rectangle = 12cm×7cm=84cm212cm \times 7cm = 84cm^2.
  2. Area of square cut-out = 3cm×3cm=9cm23cm \times 3cm = 9cm^2.
  3. Remaining Area = 84cm2−9cm2=75cm284cm^2 - 9cm^2 = 75cm^2.

Explanation:

This problem uses the 'Subtraction' method. Instead of splitting the shape, it is easier to calculate the total area of the original rectangle and subtract the area of the piece that was removed.