How to Calculate Area: Formulas for Every Shape
Area measures the amount of two-dimensional space inside a shape. This guide covers the formula for every common shape — with worked examples and the reasoning behind each formula.
What Is Area?
Area is measured in square units: cm², m², in², ft², etc. If you tile a floor with 1cm × 1cm tiles and it takes 500 tiles, the floor area is 500 cm².
Rectangle
A = l × w
The most fundamental area formula. Multiply length by width.
Example: A room 5m × 4m: A = 5 × 4 = 20 m²
Square
A = s^2
A special rectangle where all sides are equal.
Example: A square tile with 30cm sides: A = 30² = 900 cm²
Triangle
A = (1) / (2) × b × h
Half the base times the height. The height must be perpendicular to the base — not the slant side.
Example: Triangle with base 8cm, height 5cm: A = ½ × 8 × 5 = 20 cm²
Why ½? A triangle is exactly half of a rectangle with the same base and height. Draw any triangle, duplicate it, flip the copy — they always form a rectangle.
Heron's Formula (when you know all three sides)
A = √(s(s-a)(s-b)(s-c))
Where s = (a + b + c)/2 is the semi-perimeter.
Example: Triangle with sides 3, 4, 5:
- s = (3+4+5)/2 = 6
- A = √(6×3×2×1) = √36 = 6 cm²
Circle
A = π r^2
Where r is the radius (half the diameter).
Example: Circle with diameter 10cm (radius 5cm): A = π × 5² = 25π ≈ 78.54 cm²
Why πr²? Imagine cutting a circle into many thin pizza slices, then rearranging them alternating up/down into a shape approaching a rectangle. The "width" approaches πr (half the circumference) and the "height" approaches r. Area = πr × r = πr².
Ellipse
A = π × a × b
Where a and b are the semi-major and semi-minor axes.
Example: Ellipse with axes 6cm and 4cm: A = π × 3 × 2 = 6π ≈ 18.85 cm²
Trapezoid (Trapezium)
A = ((a + b)) / (2) × h
Where a and b are the parallel sides and h is the perpendicular height.
Example: Trapezoid with parallel sides 8cm and 5cm, height 4cm: A = (8+5)/2 × 4 = 6.5 × 4 = 26 cm²
Parallelogram
A = b × h
Base times perpendicular height (not the slant side).
Example: Parallelogram with base 7cm, height 3cm: A = 7 × 3 = 21 cm²
Rhombus (from diagonals)
A = (d_1 × d_2) / (2)
Where d₁ and d₂ are the two diagonals.
Example: Rhombus with diagonals 10cm and 6cm: A = (10 × 6)/2 = 30 cm²
Regular Polygon (n equal sides)
A = (1) / (4) n s^2 cot((π) / (n))
Where n = number of sides and s = side length.
Example: Regular hexagon (n=6) with side 4cm: A = ¼ × 6 × 16 × cot(π/6) = 24 × √3 ≈ 41.57 cm²
Sector of a Circle
A = (θ) / (360°) × π r^2
A "pizza slice" of a circle, where θ is the angle in degrees.
Example: Sector with radius 5cm, angle 90°: A = (90/360) × π × 25 = 25π/4 ≈ 19.63 cm²
Annulus (Ring)
A = π(R^2 - r^2)
The area between two concentric circles, where R is the outer radius and r is the inner radius.
Example: Ring with outer radius 8cm, inner radius 5cm: A = π(64 − 25) = 39π ≈ 122.52 cm²
Composite Shapes
For irregular shapes, break them into simpler pieces:
Example: An L-shaped room.
Treat it as a large rectangle minus a smaller rectangle:
- Large rectangle: 8m × 6m = 48 m²
- Missing corner: 3m × 2m = 6 m²
- L-shape area: 48 − 6 = 42 m²
Unit Conversions for Area
Since area is two-dimensional, unit conversions are squared:
| From | To | Multiply by | |------|----|------------| | 1 m² | cm² | 10,000 | | 1 ft² | in² | 144 | | 1 acre | ft² | 43,560 | | 1 hectare | m² | 10,000 | | 1 mile² | acres | 640 |
Calculate Area Now
Our shape calculators handle all the above — enter your measurements and get the area instantly with step-by-step working.