Regular Pentagon Calculator

Calculate all properties of a regular pentagon. Enter one measurement (Side, Apothem, or Circumradius) to find the Area, Perimeter, and Angles.

Formulas for a Regular Pentagon

A regular pentagon is a polygon with five equal sides and five equal internal angles ($\alpha = 108^\circ$). All calculations are based on the length of the side ($s$), the apothem ($a$), or the circumradius ($R$).

Area Calculation (A)

The standard formula uses the side length ($s$):

Alternatively, using the apothem ($a$) and perimeter ($P$):

Key Relationships (Side, Apothem, Radius)

The calculator finds the relationships using trigonometry ($\theta = 360^\circ / 10 = 36^\circ$):

• Apothem ($a$) from Side ($s$): $a = \frac{s}{2 \tan(36^\circ)}$ (approx $0.6882 \cdot s$)
• Radius ($R$) from Side ($s$): $R = \frac{s}{2 \sin(36^\circ)}$ (approx $0.85065 \cdot s$)

Irregular Pentagon Area

Our calculator focuses on the **regular** pentagon. If you are trying to find the area of an **irregular** pentagon (where sides and angles are unequal), you typically need the coordinates of all five vertices ($x_1, y_1$ through $x_5, y_5$) and must use the complex **Surveyor's Formula** (or Shoelace Formula). Simple side lengths are not enough to define an irregular pentagon.

Frequently Asked Questions (FAQ)

What is the internal angle of a regular pentagon?

The internal angle of a regular pentagon is exactly $\mathbf{108}$ degrees. The sum of all internal angles is $540^\circ$.

What is the apothem of a pentagon?

The apothem is the segment that connects the center of a regular polygon to the midpoint of one of its sides, and is perpendicular to that side. It is used to calculate the area.

How many sides does a pentagon have?

A pentagon has five sides and five vertices (corners).

','

A = \frac{1}{4} s^2 \sqrt{25 + 10\sqrt{5}}

A = \frac{1}{2} P a \quad \text{where} \quad P = 5s

$ A = \frac{1}{4} s^2 \sqrt{25 + 10\sqrt{5}} $

$ A = \frac{1}{2} P a \quad \text{where} \quad P = 5s $