Home » Math & Conversions » Geometry » Pentagon Calculator Regular Pentagon Calculator Calculate all properties of a regular pentagon. Enter one measurement (Side, Apothem, or Circumradius) to find the Area, Perimeter, and Angles. Input (Enter ONE Value) Side Length (s): Apothem (a): Circumradius (R): Calculate Properties Results Area (A) Perimeter (P) Apothem (a) Circumradius (R) Internal Angle 108° 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 ($s$): $$ A = \frac{1}{4} s^2 \sqrt{25 + 10\sqrt{5}} $$ Alternatively, using the apothem ($a$) and perimeter ($P$): $$ A = \frac{1}{2} P a \quad \text{where} \quad P = 5s $$ Key Relationships (Side, Apothem, Radius) The calculator finds the relationships using trigonometry ($\theta = 360^\circ

Subcategories in Home » Math & Conversions » Geometry » Pentagon Calculator Regular Pentagon Calculator Calculate all properties of a regular pentagon. Enter one measurement (Side, Apothem, or Circumradius) to find the Area, Perimeter, and Angles. Input (Enter ONE Value) Side Length (s): Apothem (a): Circumradius (R): Calculate Properties Results Area (A) Perimeter (P) Apothem (a) Circumradius (R) Internal Angle 108° 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 ($s$): $$ A = \frac{1}{4} s^2 \sqrt{25 + 10\sqrt{5}} $$ Alternatively, using the apothem ($a$) and perimeter ($P$): $$ A = \frac{1}{2} P a \quad \text{where} \quad P = 5s $$ 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). Related Calculators Regular Polygon Calculator Hexagon Calculator Octagon Calculator Triangle Calculator Circle Calculator Area Calculator Perimeter Calculator Solid Geometry Calculator

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