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Pentagon calculator - diagonals, area, perimeter, sides

Pentagon calculator will help you calculate the long diagonal of the pentagon, the medium diagonal of the pentagon, the shorter diagonal of the pentagon, the side length, height, area of the pentagon, circumference and radius of the circumscribed circle and the radius of the circle inscribed in the regular pentagon.

Diagonal of a regular pentagon

$$p=\frac{\sqrt{5}+1}{2}\cdot a$$

Height of a regular pentagon

$$h=\frac{\sqrt{5+2\sqrt{5}}}{2}\cdot a$$

Perimeter of a regular pentagon

$$L = 5\cdot a$$

Area of a regular pentagon

$$P_{pow} = \frac{a^2}{4}\cdot\sqrt{25+10\cdot\sqrt{5}}$$

Radius of the circle circumscribed on a regular pentagon

$$R=\frac{a\cdot\sqrt{50+10\cdot\sqrt{5}}}{10}$$

Radius of the circle inscribed in a regular pentagon

$$r=\frac{a\cdot\sqrt{25+10\cdot\sqrt{5}}}{10}$$

Pentagon - information

Regular pentagon - a convex figure, a pentagon with all sides of equal length and all angles. Regular pentagons are the walls of polyhedrons such as regular dodecahedron and truncated icosahedron.

A regular pentagon with side length a has the following properties:
1. every inside angle has a measure $${\displaystyle {\frac {3}{5}}\pi =108^{\circ }}$$
2. the center angle of the circumscribed circle based on the side of the pentagon has a measure $${\displaystyle {\frac {2}{5}}\pi =72^{\circ }}$$
3. the area of this figure is expressed by the formula $${\displaystyle P={\frac {5a^{2}}{4}}\operatorname {ctg} {\frac {\pi }{5}}={\frac {a^{2}}{4}}{\sqrt {25+10{\sqrt {5}}}}\approx 1{,}72048\cdot a^{2}}$$
4. the radius of the circle circumscribed on a regular pentagon has length $${\displaystyle R=a{\frac {\sqrt {50+10{\sqrt {5}}}}{10}}=a{\frac {1}{\sqrt {3-\varphi }}}}$$
5. the radius of the circle inscribed in a regular pentagon has the length $${\displaystyle r={\frac {a{\sqrt {25+10{\sqrt {5}}}}}{10}}={\frac {R}{\varphi }}}$$
6. the diagonal has the length $${\displaystyle d={\frac {{\sqrt {5}}+1}{2}}a=\varphi a}$$ where $$\varphi$$ is a golden number