Perimeter Calculator

The perimeter is the total distance around the outside of a 2D shape, measured by summing all side lengths. It is expressed in linear units (meters, feet, inches, etc.), not square units.

The perimeter of a rectangle is P = 2(l + w), where l is the length and w is the width. For a square (l = w), this simplifies to P = 4s, where s is the side length.

The perimeter of a circle is called the circumference: C = 2πr = πd, where r is the radius and d is the diameter. π ≈ 3.14159.

The perimeter of a triangle is P = a + b + c, where a, b, and c are the lengths of the three sides. For an equilateral triangle: P = 3s. For an isosceles triangle: P = 2a + b.

For irregular shapes, measure or sum each side length individually. For curved boundaries, use integration or approximation methods. The total perimeter is the sum of all boundary lengths.

For a regular polygon with n sides of length s: P = n × s. For example, a regular hexagon with side 5 has perimeter 30.

The perimeter of an ellipse cannot be expressed in a simple closed form. Ramanujan's approximation is: P ≈ π[3(a+b) - √((3a+b)(a+3b))], where a and b are the semi-major and semi-minor axes.

Perimeter measures the boundary length (linear units: meters, feet). Area measures the enclosed surface (square units: m², ft²). A shape can have a large perimeter but small area (e.g., a thin ring).

The perimeter of a semicircle is P = πr + 2r = r(π + 2), which includes the curved arc (πr) plus the diameter (2r). Do not forget to add the straight edge.

The perimeter of a trapezoid is P = a + b + c + d, where a and b are the parallel sides (bases) and c and d are the non-parallel sides (legs).

The perimeter of a parallelogram is P = 2(a + b), where a and b are the lengths of adjacent sides. Opposite sides are equal in a parallelogram.

The perimeter of a rhombus is P = 4s, where s is the side length. All four sides of a rhombus are equal, so you only need to know one side.

Perimeter is used for fencing yards, framing pictures, measuring room boundaries for baseboards, calculating running track distances, and determining material needed for edging or borders.

The perimeter of a circular sector is P = 2r + rθ (in radians) or P = 2r + (θ/360) × 2πr (in degrees). This includes the two radii plus the arc length.

Perimeter converts using the same factors as length. For example, 1 meter = 3.281 feet = 39.37 inches. Multiply the perimeter by the appropriate conversion factor.

For a given area, the shape with minimum perimeter is a circle. Among rectangles, the square has the minimum perimeter for a given area. This is the isoperimetric inequality.

Yes, infinitely many shapes can have the same perimeter but very different areas. A thin rectangle and a square with the same perimeter have very different areas. The circle maximizes area for a given perimeter.

The perimeter of a kite is P = 2(a + b), where a and b are the lengths of the two distinct pairs of equal adjacent sides. Unlike a parallelogram, adjacent sides (not opposite) are equal.

For a polygon with vertices (x₁,y₁), (x₂,y₂), ..., (xₙ,yₙ), the perimeter is P = Σ√((xᵢ₊₁ - xᵢ)² + (yᵢ₊₁ - yᵢ)²). This uses the distance formula between consecutive vertices.

For a regular star polygon {n/k} with n points, the perimeter is P = n × s, where s is the length of each edge (which may be different from the convex polygon side). The edge length depends on the star's geometry.

Add together the lengths of all outer edges of the composite shape. Shared interior edges are not counted — only the external boundary contributes to perimeter.

A regular hexagon has 6 equal sides. Its perimeter is P = 6s, where s is the side length. If the side is 10, the perimeter is 60 units.

Circumference is the specific term for the perimeter of a circle. All circles have circumference, but only circles use the term "circumference" — other shapes use "perimeter."

The isoperimetric inequality states that for a given area, the circle has the smallest perimeter. Equivalently, for a given perimeter, the circle encloses the maximum area. This principle guides efficient design in nature and engineering.

The Perimeter Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Perimeter Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.