Pentagon calculator
Online calculator and formulas for a regular pentagon
This function calculates various parameters of a regular pentagon, a polygon with 5 vertices. Enter one of the known parameters for the calculation.
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Properties of a regular pentagon
A pentagon is a geometric figure that belongs to the group of polygons. It is defined by five points. If all five sides are the same length, it is an equilateral pentagon. If all the angles at the five corners are the same size, the pentagon is called regular.
The sum of the interior angles of a regular pentagon is 540° (3 x 180°). This comes from a general formula for polygons, in which the number of vertices of the polygon is used as a variable \(n\):
\((n-2)· 180\ \ \ = (5-2) · 180 \ \ = 3 · 180 =540°\)
The value of an interior angle of 108° is therefore given by the formula:
\(\displaystyle \frac{180·(n-2)}{n} \ \ =\frac{180·(5-2)}{5} =108° \)
These two formulas apply to all regular polygons. The number of corners is used for n.
Formulas for the regular pentagon
Perimeter(P)
\(\displaystyle P = a · 5 \)
Area (A)
\(\displaystyle A =\frac{a^2}{4} · \sqrt{25+10 · \sqrt{5}} \ \ \) \(\displaystyle ≈\frac{a^2}{4} ·6.88191 \)
Height (h)
\(\displaystyle h = ra+ri\)
\(\displaystyle h =\frac{a}{2} · \sqrt{5 +2· \sqrt{5}} \) \(\displaystyle \ \ ≈\frac{a}{2} · 3.07768 \)
Diagonal (d) berechnen
\(\displaystyle d = \frac{a}{2} ·(1+ \sqrt{5 }) \) \(\displaystyle \ \ ≈\frac{a}{2} · 3.23607 \)
Circumference radius (rc)
\(\displaystyle rc = \frac{a}{2·cos(54)}\) \(\displaystyle \ \ ≈\frac{a}{ 1.17557}\)
Inner circle radius (ri)
\(\displaystyle ri= \sqrt{ra^2-a^2}\)
Side length (a)
\(\displaystyle a = \frac{ h · 2}{ \sqrt{5+2·\sqrt{5}}} \) \(\displaystyle \ \ ≈ \frac{ h · 2}{ 3.07768} \)
\(\displaystyle a = \sqrt{ \frac{ A · 4}{ \sqrt{25+10·\sqrt{5}}} } \) \(\displaystyle \ \ ≈ \sqrt{ \frac{ A · 4}{6.88191} } \)
\(\displaystyle a = \frac{P}{5}\)
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