Area Calculator
A free area calculator that computes the surface area of eight common 2D shapes: square, rectangle, triangle, circle, trapezoid, parallelogram, ellipse, and rhombus. Choose your shape, enter the dimensions in any consistent unit (cm, m, in, ft), and the tool shows the formula being applied, the substituted values, and the final area in square units. The calculator uses the standard geometry formulas — A = s² for a square, A = l × w for a rectangle, A = ½ × base × height for a triangle, A = π × r² for a circle, A = ½(a + b) × h for a trapezoid, and so on — and a clear formula reference table is shown below each result. Useful for school geometry homework, calculating flooring or paint coverage for a room, sizing a garden bed, working out fabric for a sewing project, or any scenario that involves a 2D measurement. Pick a shape and start typing to see the result instantly. To convert between square units (m², ft², in², acres), use our area unit converter.
Formula: A = l × w
Result
A = l × w A = 8 × 5 A = 40
Area Formulas for All Shapes
| 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 = πr² | r = radius |
| Trapezoid | A = ½(a + b) × h | a, b = parallel sides, h = height |
| Parallelogram | A = b × h | b = base, h = perpendicular height |
| Ellipse | A = πab | a = semi-major axis, b = semi-minor axis |
| Rhombus | A = (d1 × d2) / 2 | d1, d2 = diagonals |
Triangle Area in Depth
Three common methods to find triangle area:
- Base × Height: A = ½ × b × h. Height must be perpendicular to the chosen base.
- Heron's Formula (all 3 sides known): compute semi-perimeter s = (a+b+c)/2, then A = √(s(s−a)(s−b)(s−c)). Example: sides 6, 8, 10 → s=12, A=√(12×6×4×2)=√576=24.
- SAS method: A = ½ab sin(C), where C is the angle between sides a and b.
Why Parallelogram and Rectangle Share a Formula
A parallelogram can be rearranged into a rectangle by cutting a triangle from one end and attaching it to the other. Both result in A = base × height. The key: height is always the perpendicular height, not the slant side.
Similarly, a trapezoid averages its two parallel sides and multiplies by the perpendicular height. When both parallel sides are equal, it becomes a rectangle: A = s × h.
Understanding π in Circle and Ellipse Formulas
The circle formula A = πr² comes from integrating thin rings from the center outward. Each ring at radius r has circumference 2πr and width dr, giving dA = 2πr dr. Integrating from 0 to R: A = πR².
The ellipse formula A = πab extends this: an ellipse with semi-axes a and b is a stretched circle. Stretching by factor k multiplies area by k, so A = π × a × b.
Area Unit Conversions
| From | To | Multiply by |
|---|---|---|
| m² | cm² | 10,000 |
| m² | ft² | 10.764 |
| ft² | in² | 144 |
| acres | ft² | 43,560 |
| acres | m² | 4,047 |
| hectares | m² | 10,000 |
| km² | hectares | 100 |
Real-World Area Examples
- Flooring: A 15 ft × 12 ft room = 180 ft². Add 10% waste for a total of 198 ft² of flooring needed. Our Square Footage Calculator handles multi-room layouts.
- Paint: A 10 ft × 9 ft wall = 90 ft². Subtract windows (3×4=12 ft²) → 78 ft² to paint. At 350 ft²/gallon, you need roughly ¼ gallon.
- Circular pool cover: A pool with 12 ft diameter has radius 6 ft. Area = π × 36 ≈ 113 ft². Choose a cover slightly larger.
- Land area: A farm 500 m × 400 m = 200,000 m² = 20 hectares ≈ 49.4 acres. Need to convert between area units? Use our area converter.
Frequently Asked Questions
What is area and how is it measured?
Area is the amount of 2D space inside a shape, measured in square units (cm², m², ft², etc.). To find area, you typically multiply two dimensions (length and width for a rectangle) or apply a shape-specific formula. Area is additive: you can find the total area of a complex shape by breaking it into simple shapes and summing their areas.
How do you find the area of a triangle?
The standard formula is A = ½ × base × height, where height is the perpendicular distance from the base to the opposite vertex. If you know all three sides, use Heron's formula: A = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. If you know two sides and the angle between them, use A = ½ab·sin(C).
What is the area formula for a trapezoid?
A trapezoid (trapezium in UK) has two parallel sides called bases (a and b) and a height h (perpendicular distance between the bases). Area = ½(a + b) × h. This averages the two parallel sides and multiplies by height. When a = b, this simplifies to a × h, which is the parallelogram formula.
What is the area of an ellipse?
Area of an ellipse = π × a × b, where a is the semi-major axis (half the longer diameter) and b is the semi-minor axis (half the shorter diameter). When a = b, this becomes πr², the circle formula. For example, an ellipse with semi-axes 5 and 3 has area π × 5 × 3 ≈ 47.12 square units.
How do I convert between area units?
Common conversions: 1 m² = 10,000 cm² = 1,000,000 mm². 1 ft² = 144 in². 1 yd² = 9 ft². 1 acre = 43,560 ft² = 4,047 m². 1 hectare = 10,000 m² = 2.471 acres. 1 km² = 100 hectares = 1,000,000 m². When converting, note that area units scale by the square of the length conversion: 1 m = 100 cm, so 1 m² = 100² = 10,000 cm².