Area and perimeter
Pick a shape, enter the measurements, and area and perimeter appear straight away.
Measurements
Results
Fill in every measurement.
How it works
Area is measured in square units and perimeter in length units. The tool works without units, so the answer is in whatever unit you entered: type centimetres and the area is square centimetres.
The area of a circle is pi times the radius squared, and the circumference is two pi times the radius. The diameter is twice the radius.
The area of a triangle is half the base times the height, where the height is measured at a right angle to the base. If you know three sides or angles instead, that belongs in the triangle solver.
The area of an ellipse is pi times the two semi-axes multiplied. Its perimeter has no simple exact formula and is computed with Ramanujan’s approximation, which comes out slightly short. For ellipses that are nearly round the error does not show in the digits we display, but it grows as the ellipse gets longer and thinner: with semi-axes of 10 and 1 the tool shows 40.639272, while the exact perimeter is 40.639742.
The interior angle of a regular polygon is 180 degrees times the number of sides minus two, divided by the number of sides. A hexagon therefore has 120 degrees.
The formulas for the other shapes
The diagonal of a square is the side times √2, and that of a rectangle √(w² + h²). A parallelogram’s area is the base times the height and its perimeter 2(base + slanted side). The height cannot be longer than the slanted side, because it is the slanted side times the sine of the angle between the sides; equal lengths give a rectangle. The trapezium gets only an area, (a + b) / 2 × h, because the slanted sides cannot be worked out from the two parallel sides and the height.
A regular polygon is split into n isosceles triangles meeting at the centre, which gives the area n × s² / (4 × tan(180° / n)). A circular sector is the angle over 360 of the whole circle: the area is πr² × v / 360, the arc length 2πr × v / 360, and the perimeter the arc plus two radii. The angle can be at most 360 degrees.
Example
A sector with radius 5 and an angle of 72 degrees is a fifth of the circle. The area is 25π / 5 = 5π ≈ 15.707963, the arc length 2π ≈ 6.2831853 and the perimeter 6.2831853 + 10 ≈ 16.283185. A hexagon with side 2 has the area 6 × 4 / (4 × tan 30°) = 6√3 ≈ 10.392305.
The perimeter of an ellipse
The exact perimeter is 4a × E(k), where a is the longer semi-axis, k² = 1 − b²/a² and E is the complete elliptic integral of the second kind. The tool uses Ramanujan’s approximation π(a + b)(1 + 3λ² / (10 + √(4 − 3λ²))) instead, where λ = (a − b) / (a + b). NIST’s handbook of mathematical functions lists thirteen known approximations and names Ramanujan’s the most accurate. With its leading error term, which the tool leaves out, it stays within 0.023% even for the extremely elongated ellipse with semi-axes 99 and 1. Without that term the tool shows 396.01652 there against an exact 396.11074, which is 0.024% short.
Squared units
Under the SI Brochure a prefix belongs to its unit and is raised to the power with it: 1 cm² is (10⁻² m)², or 10⁻⁴ m². A rectangle of 4 by 3 metres has an area of 12 m², and the same rectangle written in centimetres, 400 by 300, has an area of 120,000 cm². Do not mix units within one shape.