Pentagon Calculator
From a single side length, get the area, the perimeter, the apothem, and the circumradius — every measurement that describes a regular five-sided polygon.
One input, four answers
Enter the side length and the calculator returns the area (¼√(5(5+2√5))s²), the perimeter (5s), the apothem (s/(2·tan36°)), and the circumradius (s/(2·sin36°)) at once.
Regular pentagons only
These formulas assume all five sides and all five angles are equal. An irregular five-sided shape needs a different method, and your answers come back in the same unit you typed in (squared for the area).
What is a pentagon calculator?
One side in, the whole pentagon out
A pentagon calculator turns a single measurement — the length of one side — into every number that describes a regular pentagon: the area it covers, the distance around it (perimeter), the apothem (centre to the middle of an edge), and the circumradius (centre to a corner). Because a regular pentagon is perfectly symmetric, all five sides are equal, so that one input fixes everything else. Pentagons show up in soccer-ball panels, the Pentagon building, home-plate in baseball, school crests, and plenty of geometry homework. Whether you are cutting a five-sided panel, sizing a logo, or finishing a worksheet, the side length is all you need.
Enter one side length in any unit to get the area, perimeter, apothem, and circumradius instantly.
A handful of short formulas, all built from the side length s and the angle 36° (a tenth of a full turn).
area = ¼ × √(5 × (5 + 2√5)) × s²A regular pentagon is five identical isosceles triangles fanned out from the centre. The perimeter is simply 5 × s. The apothem — the distance from the centre to the middle of any side — is s / (2 × tan 36°). The circumradius — the distance from the centre out to a corner — is s / (2 × sin 36°), and it is always a little longer than the apothem.
Suppose you have a regular pentagon with a side of 6.
Perimeter
5 × 6 = 30 — the distance all the way around.
Apothem and circumradius
apothem = 6 / (2 × tan 36°) = 4.129146, circumradius = 6 / (2 × sin 36°) = 5.103905.
Area
¼ × √(5 × (5 + 2√5)) × 6² = 61.937186 square units — the surface inside.
The outputs answer different practical questions. The area (about 61.937186 square units for s = 6) is how much surface the pentagon covers — the panel you would cut, the felt you would lay, the patch on a soccer ball. The perimeter (30 here) is the edging or trim that runs around it. The apothem (about 4.129146) is the centre-to-edge distance, the radius of the largest circle that fits inside; it is the natural "inner radius" if you are inscribing the shape. The circumradius (about 5.103905) is the centre-to-corner distance, the radius of the smallest circle that fits around the pentagon — exactly what you set on a compass to mark out the five corners. A handy fact: the circumradius is always larger than the apothem, because a corner is farther from the centre than the middle of a side, and dividing the area by the perimeter gives you back half the apothem.
The formulas are exact, but a couple of practical points are worth keeping in mind.
Regular pentagons and consistent units
These formulas describe a perfect regular pentagon — five equal sides and five equal 108° angles. An irregular pentagon (sides or angles that differ) will not match these results; split it into triangles instead. The side length is also unit-agnostic, so the answers are only meaningful if you keep one unit throughout: a side in centimetres gives an area in square centimetres and a circumradius in centimetres, never a mix.