Whether you're buying carpet, planning a garden, tiling a bathroom, or revising for a maths exam, knowing how to calculate area is a genuinely useful skill. This guide covers the area formula for every common shape, with worked examples, and explains how to handle irregular shapes that don't fit neatly into a single formula.
Area Formulas Quick Reference
| Shape | Formula | Variables |
|---|---|---|
| Square | A = s² | s = side length |
| Rectangle | A = l × w | l = length, w = width |
| Triangle | A = ½ × b × h | b = base, h = perpendicular height |
| Circle | A = π × r² | r = radius |
| Parallelogram | A = b × h | b = base, h = perpendicular height |
| Trapezoid/Trapezium | A = ½ × (a+b) × h | a, b = parallel sides, h = height |
| Ellipse | A = π × a × b | a, b = semi-axes |
| Regular hexagon | A = (3√3 / 2) × s² | s = side length |
| Sector (pie slice) | A = (θ/360) × π × r² | θ = angle in degrees, r = radius |
Worked Examples
Rectangle (Room flooring)
A bedroom is 4.2 m × 3.6 m. What is the floor area?
A = 4.2 × 3.6 = 15.12 m²
Add 10% for cutting waste: 15.12 × 1.10 = 16.63 m² — so order 17 m² of flooring.
Triangle (Garden bed)
A triangular garden bed has a base of 5.5 m and a perpendicular height of 3.2 m.
A = ½ × 5.5 × 3.2 = ½ × 17.6 = 8.8 m²
Important: The height must be perpendicular (at right angles) to the base, not the slant side.
Circle (Lawn sprinkler coverage)
A sprinkler covers a circular area with radius 4 m. What area does it water?
A = π × 4² = π × 16 = 50.27 m²
Trapezoid (Cross-section of a ditch)
A drainage ditch has parallel top width of 2.4 m, bottom width of 1.2 m, and depth of 1.0 m.
A = ½ × (2.4 + 1.2) × 1.0 = ½ × 3.6 × 1.0 = 1.8 m²
Heron's Formula: Triangle Without Height
If you know all three sides of a triangle (a, b, c) but not the height, use Heron's formula:
s = (a + b + c) ÷ 2 (semi-perimeter)
A = √[s(s−a)(s−b)(s−c)]
Example: Triangle with sides 6 m, 8 m, 10 m:
s = (6 + 8 + 10) ÷ 2 = 12
A = √[12 × 6 × 4 × 2] = √576 = 24 m²
(This is a 3-4-5 right triangle scaled up, so we can verify: ½ × 6 × 8 = 24 ✓)
Irregular Shapes: Decomposition Method
Most real-world spaces aren't perfect rectangles. The decomposition method works for any irregular shape:
- Divide the shape into simple sub-shapes (rectangles, triangles, etc.)
- Calculate the area of each sub-shape
- Add them all together
For an L-shaped room: treat it as two rectangles. Measure each rectangle separately and add the areas.
For an area with a circular cutout: calculate the full rectangle area, then subtract the circle area.
The Shoelace Formula: Any Polygon
For a polygon with known coordinates (x₁,y₁), (x₂,y₂), ..., (xₙ,yₙ):
A = ½ |Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|
This formula works for any polygon if you know the vertex coordinates — useful in surveying, mapping, and computer graphics where complex land parcels or plots need to be measured.
Unit Conversions for Area
| From | To | Multiply by |
|---|---|---|
| m² | cm² | 10,000 |
| m² | ft² | 10.764 |
| ft² | m² | 0.0929 |
| hectares | m² | 10,000 |
| acres | hectares | 0.4047 |
| km² | acres | 247.1 |
Summary
Rectangle: length × width. Triangle: ½ × base × height. Circle: π × r². Trapezoid: ½ × (a+b) × height. For irregular shapes, decompose into simple sub-shapes and sum their areas. Use Heron's formula when you know all three sides of a triangle but not the height. Always add 5–15% for material waste when ordering flooring, tiles, or fabric.