Area Calculator
Find the area and perimeter of rectangles, triangles, circles, sectors, trapezoids, parallelograms, ellipses, regular polygons, rings and irregular polygons from their coordinates — with a scale drawing, the working, and instant conversion to nine area units.
Written by Suhaib Hassan. Each formula is standard geometry, and the worked examples are re-derived by hand. Our methodology.
It applies the closed-form area formula for the shape you pick — Heron’s formula for three sides, the Shoelace formula for coordinates — and converts to other units exactly.
Area jobs come in many shapes and units. One page handles them all and shows the working, so you can check your own answer.
—
Area in other units
The same area expressed in nine units. Your input unit is highlighted.
| Unit | Area | Factor to m² |
|---|
Scale drawing
The shape drawn from your dimensions.
Step-by-step working
—
Area formulas reference
Every formula used on this page.
| Shape | Inputs | Area | Perimeter |
|---|---|---|---|
| Rectangle | length l, width w | A = l·w | 2(l + w) |
| Triangle | base b, height h | A = ½·b·h | a + b + c |
| Triangle (Heron) | sides a, b, c | √[s(s−a)(s−b)(s−c)] | 2s |
| Triangle (SAS) | sides a, b and angle C | ½·a·b·sin C | a + b + c |
| Circle | radius r | πr² | 2πr |
| Circular sector | radius r, angle θ | ½r²θ (θ in rad) | 2r + rθ |
| Trapezoid | parallel sides a, b; height h | ½(a + b)·h | a + b + both legs |
| Parallelogram | base b, height h | A = b·h | 2(b + slanted side) |
| Ellipse | semi-axes a, b | πab | ≈ Ramanujan II |
| Regular polygon | n sides of length s | n·s² ÷ (4·tan(π/n)) | n·s |
| Irregular polygon | vertices (xᵢ, yᵢ) | ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)| | Σ edge lengths |
| Annulus | R and r | π(R² − r²) | 2π(R + r) |
Worked examples
Flooring a room
A room is 4.2 m by 3.6 m; add 10% for waste.
with waste: 15.12 × 1.10 = 16.63 m²
Land parcel from coordinates
Corners (0,0), (100,0), (100,60), (40,90), (0,60) in metres.
A = ½ × 15000 = 7,500 m²
Triangle from three sides
Sides 13, 14 and 15.
A = √(21 · 8 · 7 · 6) = √7056 = 84
Among all shapes with the same perimeter, the circle encloses the most area (the isoperimetric inequality).
What is area?
Area measures the amount of flat surface inside a boundary, in square units: a 3 m by 4 m rectangle covers 12 m², meaning twelve 1 m by 1 m squares. Perimeter, by contrast, is the length of the boundary.
Two formulas worth knowing
- Heron’s formula gives a triangle’s area from its three sides alone: A = √[s(s − a)(s − b)(s − c)], where s is half the perimeter.
- The Shoelace formula gives the area of any simple polygon from its ordered vertices: A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. Surveyors use it for parcels defined by coordinates.
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
Sources
- Archimedes, Measurement of a Circle (c. 250 BCE), Proposition 1.
- Heron of Alexandria, Metrica (c. 60 AD) — area of a triangle from its sides.
- Braden, B. (1986). The Surveyor’s Area Formula. The College Mathematics Journal 17(4), 326–337.
- NIST Special Publication 811, Guide for the Use of the International System of Units (SI) — area units and conversion factors.
- ISO 80000-2:2019, Quantities and units — Part 2: Mathematics.
Frequently asked questions
Why is the area of a circle πr² and not something with the diameter?
Area scales with the square of length, and the circle’s size is set by one length, the radius. Archimedes showed that a circle has the same area as a right triangle whose legs are the radius and the circumference: ½ × r × 2πr = πr². In terms of the diameter it is π d² ÷ 4.
How does the Shoelace formula work, and does vertex order matter?
It sums the cross products of consecutive vertex coordinates around the boundary. Vertices must be listed in order around the edge (clockwise or anticlockwise); the absolute value handles either direction. If edges cross each other, the formula no longer gives the enclosed area.
How do I convert between square metres, square feet and acres?
Convert the length factor and square it: 1 ft = 0.3048 m, so 1 ft² = 0.09290304 m², and 1 m² ≈ 10.7639 ft². An acre is 4,046.8564224 m² (43,560 international ft²), and a hectare is 10,000 m². The table above does all of these at once.
Is the ellipse perimeter exact?
The area πab is exact. The perimeter of an ellipse has no simple closed form, so this calculator uses Ramanujan’s second approximation, which is accurate to well under one part in a million for ordinary ellipses.
Why do trapezoid and parallelogram perimeters need extra sides?
Their areas depend only on the base(s) and the perpendicular height, but the slanted sides can have different lengths for the same height, so the perimeter cannot be found from those inputs alone.
Which units should I use for the inputs?
Any length unit, as long as every dimension uses the same one. The area comes out in that unit squared, and the conversion table shows it in other area units.
Related calculators
Continue with geometry.