Area Formula Chart
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | length × width | 2(l + w) |
| Square | side² | 4 × side |
| Circle | πr² | 2πr |
| Triangle | ½ × base × height | a + b + c |
| Trapezoid | ½(a + b) × height | sum of the sides |
| Parallelogram | base × height | 2(a + b) |
| Ellipse | πab | approximation only |
| Sector | πr² × angle/360 | 2r + arc length |
Heron's Formula
When only the three sides of a triangle are known, the height is not needed:
s = (a + b + c) / 2 Area = √(s(s−a)(s−b)(s−c))
It works for any triangle and is roughly two thousand years old. Note the triangle inequality it depends on: no side may be longer than the other two combined, or the expression under the root goes negative.
Height Means Perpendicular Height
For triangles, parallelograms and trapezoids, the height is the perpendicular distance between the base and the opposite side or vertex — not the length of a slanted edge. Using the slant length is the single most common error in area calculations, and it always overstates the answer.
Area Scales With the Square
Doubling every dimension quadruples the area; tripling them multiplies it by nine. This has practical consequences: a 16-inch pizza has 2.56 times the area of a 10-inch, and a room twice as long and twice as wide needs four times the flooring.
Unit Conversions
| From | To | Multiply by |
|---|---|---|
| Square metres | Square feet | 10.7639 |
| Square feet | Square metres | 0.092903 |
| Square feet | Acres | 1 / 43,560 |
| Square metres | Hectares | 1 / 10,000 |
| Acres | Hectares | 0.404686 |
Note that converting area between unit systems uses the square of the linear factor: one foot is 0.3048 m, so one square foot is 0.3048² = 0.092903 m².
Frequently Asked Questions
How do I find the area of an irregular shape?
Divide it into rectangles and triangles, compute each, and add them. For genuinely irregular outlines — a lake, a plot of land — the surveyor's shoelace formula works from coordinates.
Why is a triangle's area half the rectangle's?
Two identical triangles fit together to form a parallelogram with the same base and height. The triangle is therefore exactly half of it.
Is there an exact formula for an ellipse's perimeter?
No elementary one exists — it requires an elliptic integral. Ramanujan's approximation, used here, is accurate to better than one part in ten thousand for most ellipses.