What is the difference between perimeter and circumference?

Perimeter is the general term for the total boundary length of any flat shape. Circumference is the specific word for the perimeter of a circle or ellipse. Every circumference is a perimeter, but only closed curves have a circumference — a rectangle has a perimeter, not a circumference.

How is the circumference of a circle calculated?

The formula is C = 2πr, where r is the radius and π ≈ 3.14159. If you know the diameter d instead, use C = πd because d = 2r. For example, a circle with radius 5 cm has a circumference of 2 × π × 5 ≈ 31.416 cm.

Why does the ellipse use an approximation?

There is no simple closed-form formula for the exact perimeter of an ellipse — it involves an infinite series called an elliptic integral. This calculator uses Ramanujan's second approximation: P ≈ π(a+b)[1 + 3h/(10 + √(4−3h))], where h = (a−b)²/(a+b)². This is accurate to within 0.00001% for typical shapes.

What is the perimeter of a sector?

A circular sector (pizza-slice shape) has two straight radii and one curved arc. The arc length for an angle θ (in radians) is rθ, so the full perimeter is P = rθ + 2r. The calculator converts your degree input to radians automatically using θ(rad) = θ(°) × π/180.

What is an annulus and how is its perimeter found?

An annulus is the region between two concentric circles — like a washer or a ring. Its boundary consists of two circles (outer radius R, inner radius r), so the total perimeter is the sum of both circumferences P = 2πR + 2πr = 2π(R + r).

How do I find the perimeter of a regular polygon?

A regular polygon has n equal sides each of length s, so P = n × s. A regular hexagon (n = 6) with side 5 cm has a perimeter of 30 cm. This calculator accepts any n ≥ 3, covering triangles, squares, pentagons, hexagons, octagons, and beyond.

Perimeter Calculator

The perimeter of a flat shape is the total distance around its outer boundary — measured in the same length unit as the sides. This calculator covers 14 common shapes, applies the correct formula for each, and shows you the full substituted working so you can check or reproduce the result by hand.

Shapes covered

Shape	Formula
Rectangle	P = 2(w + h)
Square	P = 4s
Equilateral triangle	P = 3s
Isosceles triangle	P = 2a + b
Scalene triangle	P = a + b + c
Circle	C = 2πr
Semicircle	P = r(π + 2)
Ellipse	P ≈ π(a+b)[1 + 3h/(10 + √(4−3h))]
Trapezoid	P = a + b + c + d
Parallelogram	P = 2(a + b)
Rhombus	P = 4s
Regular n-gon	P = n · s
Circular sector	P = rθ + 2r
Annulus	P = 2π(R + r)

How it works

Each formula is derived from first principles. For straight-sided shapes, you add up every side. For shapes involving curves, arc length comes from the relationship arc = radius × angle in radians, which is the definition of the radian itself (one radian is the angle subtended by an arc equal in length to the radius).

The ellipse is the one shape that has no exact closed-form expression — only an infinite series. Ramanujan’s approximation (1914) is used here because it is accurate to better than 0.00001% for all practical ellipses:

h = (a−b)² / (a+b)²

P ≈ π(a+b) × [1 + 3h / (10 + √(4−3h))]

When a = b the ellipse becomes a circle: h = 0 and the formula collapses to 2πa, exactly matching C = 2πr. Every other formula is exact (within floating-point precision).

Worked examples

Rectangle 10 cm × 5 cm

P = 2(10 + 5) = 2 × 15 = 30 cm

Circle, radius 7 m

C = 2π × 7 = 43.982 m (4 d.p.)

Regular hexagon, side 5 cm

P = 6 × 5 = 30 cm

Circular sector, radius 5 cm, angle 60°

Convert: 60° × π/180 = 1.0472 rad. Arc = 5 × 1.0472 = 5.236 cm. P = 5.236 + 2 × 5 = 15.236 cm

Annulus, outer radius 8 cm, inner radius 5 cm

P = 2π(8 + 5) = 2π × 13 = 81.681 cm

A note on units

The calculator returns the perimeter in the unit you entered — it does not convert between unit systems. If your dimensions are in centimetres, your answer is in centimetres. To convert: 1 m = 100 cm, 1 in = 2.54 cm, 1 ft = 30.48 cm. Make sure all inputs use the same unit before calculating.