Regular Polygon Calculator

Regular Polygon Calculator: Find Side Length, Area, Perimeter, and More

A regular polygon is a polygon that is both equiangular and equilateral. All sides are equal in length and all interior angles are equal. Our regular polygon calculator lets you compute all important properties of any regular polygon from a single input value. Simply choose the number of sides, enter what you know (side length, inradius, circumradius, area, or perimeter), and the calculator instantly determines everything else.

What is a Regular Polygon?

A regular polygon is a symmetric flat shape with straight sides of equal length and equal angles. Common examples include equilateral triangles (3 sides), squares (4 sides), pentagons (5 sides), hexagons (6 sides), octagons (8 sides), and decagons (10 sides). Regular polygons can have any number of sides from 3 up to thousands.

The center of a regular polygon is equidistant from all vertices (circumcenter) and from all side midpoints (incenter). The distance from the center to a vertex is called the circumradius (R), and the distance from the center to the midpoint of a side is called the inradius or apothem (r).

Key Properties of Regular Polygons

A regular polygon has several measurable properties that are mathematically related. When you know the number of sides and any one of these values, all others can be calculated:

Side Length (a): The length of each equal side of the polygon.
Inradius / Apothem (r): The distance from the center to the midpoint of any side. It is the radius of the inscribed circle.
Circumradius (R): The distance from the center to any vertex. It is the radius of the circumscribed circle.
Area (A): The total area enclosed by the polygon.
Perimeter (P): The total distance around the polygon, equal to n times the side length.
Interior Angle (x): The angle between two adjacent sides inside the polygon.
Exterior Angle (y): The angle between a side and an extended adjacent side. Always equals 360/n.

How to Use This Calculator

Using the regular polygon calculator is straightforward:

Select the number of sides: Choose a common polygon from the dropdown, or enter a custom value up to 1000 sides.
Choose what you know: Pick the known value type from the dropdown — side length, inradius, circumradius, area, or perimeter.
Enter the value: Type in your known measurement.
View all results: All other properties are calculated and displayed instantly in real time.

Formulas for Regular Polygon Calculations

The following formulas are used to compute all regular polygon properties, where n is the number of sides, a is the side length, r is the inradius, and R is the circumradius:

Side Length: a = 2r tan(π/n) = 2R sin(π/n)
Inradius: r = (1/2)a cot(π/n) = R cos(π/n)
Circumradius: R = (1/2)a csc(π/n) = r sec(π/n)
Area: A = (1/4)na² cot(π/n) = nr² tan(π/n)
Perimeter: P = na
Interior Angle: x = ((n-2)/n) × 180°
Exterior Angle: y = 360° / n
Incircle Area: A_ic = πr²
Circumcircle Area: A_cc = πR²

Applications of Regular Polygons

Regular polygons appear throughout mathematics, engineering, architecture, and nature. Hexagons are found in honeycomb structures for their efficient packing properties. Octagons are common in architecture, particularly for stop signs and building designs. Pentagon shapes appear in government buildings and molecular structures. Engineers use polygon calculations for gear design, tile layout, and structural analysis. Understanding regular polygon properties is fundamental to geometry, trigonometry, and many practical design applications.

Also check: Triangle Calculator, Square Calculator, Circle Calculator, Area Calculator, Volume Calculator, and Surface Area Calculator.

Frequently Asked Questions

What is the difference between inradius and circumradius?

The inradius (r), also called the apothem, is the distance from the center of the regular polygon to the midpoint of any side. It is the radius of the inscribed circle that touches each side. The circumradius (R) is the distance from the center to any vertex. It is the radius of the circumscribed circle that passes through all vertices. For any regular polygon, the circumradius is always larger than the inradius.

Can I calculate properties for polygons with more than 12 sides?

Yes, our calculator supports regular polygons with any number of sides from 3 to 1000. When you select a value above 12 from the dropdown, a custom input field appears where you can enter any number between 3 and 1000. This allows you to calculate properties for tridecagons (13 sides), tetradecagons (14 sides), or any regular polygon up to a 1000-gon.

How do I find the area of a regular polygon if I only know the side length?

If you know the side length (a) and the number of sides (n), the area is calculated using the formula A = (1/4)na² cot(π/n). Simply select "Side Length (a)" as your input mode, enter the side length, and choose the number of sides. The area will be calculated instantly along with all other properties.

What is the interior angle of a regular pentagon?

A regular pentagon has 5 sides. Using the formula x = ((n-2)/n) × 180°, the interior angle is ((5-2)/5) × 180° = (3/5) × 180° = 108°. Each interior angle of a regular pentagon measures 108 degrees.

How does the number of sides affect the shape of a regular polygon?

As the number of sides increases, a regular polygon becomes increasingly round. The interior angle approaches 180° (a straight line), and the exterior angle approaches 0°. With very many sides, the polygon closely approximates a circle. For example, a regular 100-gon has interior angles of 176.4° and is virtually indistinguishable from a circle to the naked eye.

What is the apothem of a regular polygon?

The apothem is another name for the inradius of a regular polygon. It is the distance from the geometric center to the midpoint of any side. The apothem is perpendicular to the side and is the radius of the inscribed circle. It plays a key role in area calculations: the area of any regular polygon can also be computed as A = (1/2) × perimeter × apothem.

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