Area Calculator

How This Calculator Works

Area is the amount of two-dimensional space a shape covers, and each common shape has its own formula for finding it from a few basic measurements. Pick a shape — rectangle, square, circle, triangle, or trapezoid — enter its dimensions, and this calculator returns the area along with the perimeter (or circumference, for a circle) where that measurement makes sense.

Every shape below reduces to one of two ideas: multiplying two lengths together (a rectangle, square, or parallelogram-shaped triangle-double), or the circle’s constant relationship between radius and the space it encloses.

The Formula

Each shape has its own formula:

$$\text{Rectangle: Area} = \vA{\text{Length}} \times \vB{\text{Width}}$$ $$\text{Square: Area} = \vA{\text{Side}}^2$$ $$\text{Circle: Area} = \pi \times \vA{\text{Radius}}^2$$ $$\text{Triangle: Area} = \frac{1}{2} \times \vA{\text{Base}} \times \vB{\text{Height}}$$ $$\text{Trapezoid: Area} = \frac{1}{2} \times (\vA{a} + \vB{b}) \times \vC{\text{Height}}$$

Worked Example

A trapezoid with parallel sides of 4 m and 6 m, and a height of 3 m:

1. Add the two parallel sides: $\vA{4} + \vB{6} = 10 \text{ m}$.
2. Multiply by the height: $10 \times \vC{3} = 30$.
3. Take half of that: $30 \div 2 = 15 \text{ m}^2$.

The same trapezoid’s shape — but not its area — also works out for a rectangle 5 m long and 3 m wide ($5 \times 3 = 15 \text{ m}^2$ too), a useful sanity check for why the trapezoid formula includes that final “divide by half”: it’s averaging the two parallel sides into an equivalent rectangle width.