Area of Compound Shapes

A compound shape is made up of two or more simple shapes joined together. To find the total area, split the compound shape into rectangles, triangles, or other familiar shapes, calculate each area separately, and add them up.

What is a Compound Shape?

A compound shape (also called a composite shape) is formed by combining two or more basic shapes such as rectangles, triangles, semicircles, or trapezoids.

Strategy: Split, Calculate, Add

Split the shape into familiar pieces (draw dotted lines).
Label all dimensions — sometimes you need to work out missing lengths first.
Calculate each piece's area using the correct formula.
Add all the areas together.

L-shaped room

An L-shape can be split into two rectangles.

Overall width: 8 m, height: 6 m; a rectangular piece 3 m × 4 m is cut away from the top-right.

Rectangle A: $8 \times 6 = 48$ m² Rectangle B (cut away): $3 \times 4 = 12$ m²

Alternatively, split into two smaller rectangles directly:

Left part: $5 \times 6 = 30$ m²
Right part: $3 \times 2 = 6$ m²
Total: $30 + 6 = 36$ m²

Either method gives the same answer.

Shapes with Semicircles

If a shape includes a semicircle, use: $\text{Semicircle area} = \frac{\pi r^2}{2}$

Rectangle with a semicircle on top

Rectangle 10 cm × 6 cm, with a semicircle of diameter 10 cm on top.

Rectangle: $10 \times 6 = 60$ cm²

Semicircle (radius = 5): $\frac{\pi \times 25}{2} \approx 39.3$ cm²

Total ≈ $99.3$ cm²

Tip

When finding the perimeter of a compound shape, only count the outer edges — not the internal dividing lines.

Real world: designing a garden with a patio

A garden is rectangular, 10 m × 8 m, with a square patio area of 3 m × 3 m in one corner that will be paved rather than turfed.

Grass area = full rectangle − patio square:

$A = (10 \times 8) - (3 \times 3) = 80 - 9 = 71 \text{ m}^2$

71 m² of turf is needed. The patio requires 9 m² of paving slabs.

Real world: L-shaped flat floor plan

A flat has an L-shaped floor plan. The outer rectangle measures 8 m × 6 m, but a 2 m × 3 m storage alcove is cut from one corner.

Full rectangle: $8 \times 6 = 48 \text{ m}^2$

Alcove removed: $2 \times 3 = 6 \text{ m}^2$

Usable floor area: $48 - 6 = 42 \text{ m}^2$

Estate agents list the property as 42 m²; a flooring contractor orders exactly 42 m² of laminate, avoiding costly over-ordering.

Real world: athletics running track

An indoor running track is a rectangle 60 m × 30 m with a semicircle on each short end (diameter 30 m, radius 15 m).

Rectangle: $60 \times 30 = 1800 \text{ m}^2$

Two semicircles = one full circle: $\pi \times 15^2 \approx 707 \text{ m}^2$

Total area $\approx 1800 + 707 = 2507 \text{ m}^2$

Venue managers use this to calculate the cost of resurfacing the track or the weight of rubber-crumb material needed.

Tip

Where do we see compound shapes? Floor plans of almost every building are compound shapes — rooms joined together create L-shapes, T-shapes, and more complex outlines. Swimming pools with curved ends, sports fields with rounded corners, and park layouts are all real-world compound shapes.

Practice Problems

A shape is made of a 10 cm × 4 cm rectangle with a triangle of base 10 cm and height 3 cm on top. Find the total area.

Show answer

Rectangle: 10 × 4 = 40 cm². Triangle: ½ × 10 × 3 = 15 cm². Total = 55 cm².

An L-shape has outer dimensions 8 m × 7 m. A 3 m × 4 m piece is removed from one corner. What is the area?

Show answer

Full rectangle: 8 × 7 = 56 m². Removed piece: 3 × 4 = 12 m². Area = 56 − 12 = 44 m².

A running track is a rectangle 60 m × 30 m with semicircles on each short end (diameter 30 m). Find the total area inside the track.

Show answer

Rectangle: 60 × 30 = 1800 m². Two semicircles = one full circle: π × 15² ≈ 707 m². Total ≈ 2507 m².