Area of Complex Shapes

Complex (composite) shapes can be broken into simple rectangles, triangles, and circles. Their areas are found by adding or subtracting the areas of these parts.

Review: Basic Area Formulas

Rectangle

A = l \times w

l = length, w = width.

Triangle

A = \frac{1}{2} \times b \times h

b = base, h = perpendicular height.

Circle

A = \pi r^2

r = radius, π ≈ 3.14159.

Strategy: Decompose and Combine

Break the shape into simpler parts whose areas you know, then add them together.

1
Draw lines to divide the shape
Split into rectangles, triangles, or semicircles.
2
Label all dimensions
Use given measurements to fill in missing lengths.
3
Calculate each part
Use the correct formula for each sub-shape.
4
Add the parts
Total area = sum of all parts.

Strategy: Subtract

Sometimes it is easier to find a large simple area and subtract cut-out sections.

$A_{\text{composite}} = A_{\text{large}} - A_{\text{cut-out}}$

Example: L-Shaped Room

An L-shaped room can be split into two rectangles. Suppose:

Top part: 4 m × 2 m = 8 m²
Bottom part: 6 m × 3 m = 18 m²
Total: 26 m²

Example: Circle Inside a Square

Square with side 10 cm, circle of radius 4 cm cut from the centre:

$A = 10^2 - \pi \times 4^2 = 100 - 50.27 \approx 49.73 \text{ cm}^2$

Tip Sketch the shape first, label all known dimensions, identify which simple shapes you can spot.

Find the area of a shape that is a rectangle 8 cm × 5 cm with a triangular notch of base 4 cm and height 3 cm removed.

$A = 8 \times 5 - \frac{1}{2} \times 4 \times 3 = 40 - 6 = 34 \text{ cm}^2$

Real world: tiling a kitchen floor

A kitchen floor is L-shaped: a 5 m × 4 m section plus a 2 m × 3 m alcove. Total area = $(5 \times 4) + (2 \times 3) = 20 + 6 = 26\text{ m}^2$ . A tiler needs to order at least this many square metres of tiles (plus a little extra for cutting).

Real world: painting a wall with a window

A wall is 5 m wide and 3 m tall but has a window 1.2 m × 1.5 m. Paintable area = $5 \times 3 - 1.2 \times 1.5 = 15 - 1.8 = 13.2\text{ m}^2$ . Knowing the composite area prevents buying too much or too little paint.

Tip Always sketch the shape before calculating. Marking where you will split it (or subtract) saves time and avoids errors.

Practice Problems

A running track is formed by a rectangle 100 m × 60 m with semicircles on the short ends (radius = 30 m). Find the total area.

Show answer

Rectangle: 100×60 = 6000 m². Two semicircles = one full circle: π×30² ≈ 2827 m². Total ≈ 8827 m².

An L-shape: outer rectangle 10×8, inner rectangle 6×5 removed from top right. Find the area.

Show answer

10×8 − 6×5 = 80 − 30 = 50 square units.