How this calculator works
A regular pentagon has five equal sides and five interior angles of 108 degrees. It can be divided into five identical isosceles triangles from the centre, which gives the area formula and the apothem. The apothem is the distance from the centre to a side, and the circumradius reaches a vertex. The diagonal of a regular pentagon is famously in the golden ratio to its side, so it equals about 1.618 times the side length. Entering the area, perimeter, a radius, the diagonal or the height first recovers the side and then builds the other values.
Formula
Side a. Perimeter = 5a. Area = (1/4)√(5(5 + 2√5))a². Apothem = a/(2 tan 36°). Circumradius = a/(2 sin 36°). Diagonal = φa, where φ = (1 + √5)/2. Height = a[1/(2 tan 36°) + 1/(2 sin 36°)].
Worked example
A pentagon with side 10 m has perimeter 50 m, area 25√(5(5 + 2√5)) ≈ 172.0477 m², apothem 5/tan 36° ≈ 6.8819 m, circumradius 5/sin 36° ≈ 8.5065 m, diagonal 10φ ≈ 16.1803 m and height ≈ 15.3884 m.
Advertisement
Frequently asked questions
Why is the diagonal tied to the golden ratio?
In a regular pentagon the ratio of a diagonal to a side is the golden ratio, about 1.618. This relationship is one of the oldest known appearances of the number.
What is the difference between the apothem and the circumradius?
The apothem goes from the centre to the midpoint of a side and is the shorter distance. The circumradius goes from the centre to a vertex and is longer.
How many diagonals does a pentagon have?
Five. A polygon with n sides has n(n - 3)/2 diagonals, which gives 5 for a pentagon. In a regular pentagon all five have the same length.
Can a pentagon tile the plane?
Not by itself with regular pentagons. Regular pentagons leave gaps when placed edge to edge, although other pentagon shapes can tile the plane.