## Properties of a Pentagon

This tool calculates the basic geometric properties of a regular pentagon. Regular polygons are equilateral (all sides equal). The tool can calculate the properties of the pentagon, given the length of its sides, or the inradius or the circumradius. Enter below the shape dimensions. The calculated results will have the same units as your input. Please use consistent units for any input.

Known data: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Geometric properties: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Area = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Perimeter = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Side α = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inradius R _{i} = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Circumradius R _{c} = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Angles : | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Interior φ = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Central θ = |

## Definitions

### Geometry

Pentagon is a polygon with five sides and five vertices. A pentagon is regular when all its sides are equal. The interior angles of a regular pentagon are always 108°. For any regular polygon, a circle that passes through all vertices can be drawn. That is the cirmuscribed circle or circumcircle of the polygon. Also, a circle that is tangent to all sides can be drawn, which is called inscribed circle or incircle.

The radius of circumcircle R_{c} and incircle R_{i} (usually called circumradius and inradius respectively), are related to the side length α and also among each other. These relationships can be discovered using the properties of the right triangle, highlighted in the figure below, leading to the following formulas:

\[ \begin{split} R_c & = \frac{a}{2 \sin{\frac{\theta}{2}}} \\ R_i & = \frac{a}{2 \tan{\frac{\theta}{2}}} \\ R_i & = R_c \cos{\frac{\theta}{2}} \end{split} \]

where θ the central angle and α the side length. For the regular pentagon, θ = 72°, therefore the above formulas are simplified to:

\[ \begin{split} R_c & = \frac{a}{2 \sin{36^{\circ}}} \approx 0.85 a \\ R_i & = \frac{a}{2 \tan{36^{\circ}}} \approx 0.69 a \\ \\ R_i & = R_c \cos{36^{\circ}} \approx 0.81 R_c \end{split} \]

The area of any regular N-sided polygon can be found, considering that the whole shape can be decomposed to N number of identical isosceles triangles (sides α,R_{c} ,R_{c} ). The area of each triangle is \(\frac{1}{2}a R_i\). Therefore for the polygon:

\[ A = \frac{N}{2} a R_i = \frac{N a^2}{4\tan{\frac{\theta}{2}}} \]

Substituting for the specific case of regular pentagon, the above formula becomes:

\[ A = \frac{5 a^2}{4 \tan{36^{\circ}}} \approx 1.72 a^2 \]

The perimeter of any N-sided regular polygon is simply the sum of the lengths of all sides: \( P = N a \). Therefore, for the regular pentagon :

\[ P = 5a \]

NOTE: Some of the above formulas have general scope for all regular polygons. These are the formulas, with θ, φ or N variables, not substituted with specific values.