Area Calculator
Free area calculator for 11 shapes, from squares and circles to trapezoids, sectors and regular polygons, with the formula, worked steps and a labelled diagram.
All lengths use the chosen unit. Decimals such as 2.5 are accepted; the area is given in square units.
- Area
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- Perimeter
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Step by step
Area in other units
| Unit | Area |
|---|
How to use the area calculator
- Pick the shape in Shape: Square, Rectangle, Circle, Triangle (base and height), Triangle (three sides), Trapezoid, Ellipse, Parallelogram, Rhombus, Circular sector or Regular polygon.
- Choose the Length unit your measurements are in: millimetres, centimetres, metres, kilometres, inches, feet, yards or miles.
- Type the dimensions in the fields that appear. Each field shows the letter used in the formula, such as Radius (r) or Diagonal 1 (d₁). Decimals like 2.5 are accepted.
- The Area and the Perimeter update as you type; Calculate does the same. Clear empties the fields and Copy copies the area.
- Read Step by step to see the formula, the formula with your values substituted and the result. A diagram of the shape, with its dimensions labelled, is drawn below the results.
- Open Area in other units to read the same area in mm², cm², m², km², in², ft², yd², mi², hectares and acres.
The result is exact whenever possible. A circle of radius 5 has an area of 25π, shown together with its decimal value ≈ 78.54. A triangle with three sides of 2 gives √3, and a regular hexagon with 4 cm sides gives 24√3 cm².
Area formulas used
| Shape | Area | Perimeter |
|---|---|---|
| Square (side a) | A = a² | P = 4a |
| Rectangle (a × b) | A = a × b | P = 2(a + b) |
| Circle (radius r) | A = π × r² | P = 2πr |
| Triangle (base b, height h) | A = b × h / 2 | needs the three sides |
| Triangle (sides a, b, c) | A = √(s(s − a)(s − b)(s − c)), s = (a + b + c) / 2 | P = a + b + c |
| Trapezoid (bases a and b, height h) | A = (a + b) / 2 × h | needs the slanted sides |
| Ellipse (semi-axes a and b) | A = π × a × b | approximation |
| Parallelogram (base b, height h) | A = b × h | needs the slanted side |
| Rhombus (diagonals d₁, d₂) | A = d₁ × d₂ / 2 | P = 2√(d₁² + d₂²) |
| Circular sector (radius r, angle θ) | A = θ / 360 × π × r² | P = 2r + arc |
| Regular polygon (n sides of length s) | A = n × s² / (4 × tan(π/n)) | P = n × s |
The height of a triangle, trapezoid or parallelogram is always measured perpendicular to the base, not along a slanted side. The diagram shows it as a dashed line with a right-angle mark.
For a triangle where you know the three sides but not the height, choose Triangle (three sides): the tool applies Heron’s formula and checks that the sides can actually form a triangle (each one must be shorter than the sum of the other two).
The perimeter of an ellipse has no simple closed formula. The tool uses Ramanujan’s second approximation, P ≈ π(a + b)(1 + 3h / (10 + √(4 − 3h))) with h = (a − b)² / (a + b)², which stays within 0.005% of the true value even for a very flat ellipse and is far more precise for usual shapes.
Square units and conversions
Area is measured in square units: if the lengths are in centimetres, the area is in cm². Converting areas is where most mistakes happen, because the conversion factor is squared. One metre is 100 cm, so one square metre is 100 × 100 = 10,000 cm². Likewise one square foot is 144 square inches, and one square yard is 9 square feet. The conversion table does this for you, and adds hectares (10,000 m²) and acres (about 4,047 m²) for land.
Examples
- A room of 4.2 m by 3.5 m: Rectangle, 14.7 m² of floor.
- A round table of 60 cm radius: Circle, 3600π ≈ 11,309.73 cm², or about 1.13 m².
- A slice of pizza cut at 45° from a 15 cm radius: Circular sector, 28.125π ≈ 88.36 cm².
- A stop sign, octagon with 25 cm sides: Regular polygon with n = 8, about 3,017.77 cm².
Frequently asked questions
What is the difference between area and perimeter?
Area measures the surface inside a shape, in square units. Perimeter measures the length of its outline, in ordinary units. Two shapes can have the same perimeter and very different areas.
Why is the perimeter not shown for some shapes?
For a triangle given by base and height, a trapezoid or a parallelogram, the base and height fix the area but not the length of the slanted sides, so the perimeter cannot be known. Use Triangle (three sides) if you know all three sides.
Why does the result contain π or a square root?
The tool keeps the exact value when it can, such as 25π or 24√3, which is what a maths exercise usually expects. The decimal value after the ≈ sign is the one to use for practical measurements.
How do I find the area of an irregular shape?
Split it into simple shapes (rectangles, triangles, sectors), calculate each area with this tool in the same unit, then add them up. Subtract the area of any hole.
Is the diagram drawn to scale?
Yes, as long as the proportions stay reasonable. Very long and thin shapes are drawn with their ratio limited to 4 : 1 so the labels remain readable; the numbers are always calculated from your real dimensions.
Related tools
- Area Converter Convert area between square metres, square feet, acres, hectares, km² and square miles, using exact definitions and the formula for each pair.
- Triangle Calculator Free triangle calculator that solves any triangle from three values (SSS, SAS, ASA, AAS or SSA), with area, perimeter, heights, steps and a scaled diagram.
- Pythagorean Theorem Calculator Pythagorean theorem calculator that finds the hypotenuse or a missing leg of a right triangle, with exact radicals, step-by-step working and a diagram.