Perimeter calculator

Pick the shape under Shape, then type its measurements separated by commas or spaces. Units only changes the label; the numbers are worked the same way.

Rectangle: the length and width, like 5, 3. Square: one side. Triangle: the three sides, like 3, 4, 5. Circle: the radius, or the diameter written as d = 10.

Regular polygon: the number of sides, then the side length, like 6, 4 for a hexagon with sides of 4. If you know the distance from the centre to a corner instead, write it as R, like n = 6, R = 4.

Trapezoid: the two parallel sides first, then the two legs, like 10, 6, 5, 5. Polygon from corners: each corner as an x, y pair in order around the shape, like (0, 0) (4, 0) (4, 3).

Perimeter is the distance around a two-dimensional shape. For any polygon it is the sum of the side lengths, and the formulas for particular shapes are shortcuts for that sum.

Rectangle: P = 2(l + w), two lengths and two widths. Square: P = 4s. Triangle: P = a + b + c. Regular polygon with n sides of length s: P = ns. Measured from the centre to a corner, R, each side is 2R sin(π/n), so P = 2nR sin(π/n).

A circle's perimeter is called its circumference: C = 2πr, or πd. The answer is given exactly, as 10π, and as a decimal. The circle calculator works the area, arcs and sectors from the same radius.

Each side's length is the distance between two neighbouring corners, from the distance formula: √((x₂ − x₁)² + (y₂ − y₁)²), which is Pythagoras applied to the horizontal and vertical gaps. The last corner joins back to the first, so don't repeat the starting point.

When every corner has whole-number coordinates, the sides are square roots of whole numbers, and the calculator collects them into an exact answer such as 4√2 alongside the decimal. The distance formula calculator shows the working for a single pair of points.

Three lengths make a triangle only if each pair adds up to more than the third. 3, 4 and 8 can't close, because 3 + 4 is less than 8, and the calculator says so rather than giving a perimeter for a shape that can't exist.

A trapezoid has a similar rule. With the parallel sides a and b and the legs c and d, the shape exists only when |d − c| < |b − a| < c + d. If the parallel sides are equal it is a parallelogram, and the legs must be equal too.

Trapezoid is the American word. In British English the same shape, a quadrilateral with a pair of parallel sides, is a trapezium.