# Regular Polygon Calculator

Calculate area, perimeter, interior angle, apothem, and circumradius of any regular polygon. Enter number of sides and side length.

## What this calculates

Compute the area, perimeter, interior angles, apothem, and circumradius of any regular polygon. Enter the number of sides (3 to 100) and the side length to get instant results.

## Inputs

- **Number of Sides (n)** — min 3, max 100 — The number of equal sides (3 to 100).
- **Side Length (s)** — min 0 — The length of each side.

## Outputs

- **Area** — The total area of the regular polygon.
- **Perimeter** — The total perimeter (n × s).
- **Interior Angle** — The measure of each interior angle.
- **Apothem** — The distance from the center to the midpoint of a side.
- **Circumradius** — The distance from the center to a vertex.

## Details

A regular polygon has all sides equal and all interior angles equal. Common examples include the equilateral triangle (3 sides), square (4), regular pentagon (5), regular hexagon (6), and regular octagon (8). As the number of sides increases, the polygon approaches a circle.

The key formulas for a regular polygon with n sides of length s are: Area = (n * s²) / (4 * tan(π/n)), Perimeter = n * s, Interior angle = (n - 2) * 180° / n, Apothem = s / (2 * tan(π/n)), and Circumradius = s / (2 * sin(π/n)).

The apothem is the perpendicular distance from the center to the midpoint of any side. The circumradius is the distance from the center to any vertex. These two measurements define the inscribed circle (touching the sides) and circumscribed circle (passing through the vertices), respectively. The area can also be computed as A = (1/2) * perimeter * apothem.

## Frequently Asked Questions

**Q: What is a regular polygon?**

A: A regular polygon is a polygon with all sides of equal length and all interior angles of equal measure. Examples include equilateral triangles, squares, regular pentagons, and regular hexagons. Irregular polygons have unequal sides or angles.

**Q: What is the apothem?**

A: The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It equals s / (2 * tan(π/n)) where s is the side length and n is the number of sides. The apothem is also the radius of the inscribed circle.

**Q: How do I find the interior angle of a regular polygon?**

A: The interior angle of a regular polygon with n sides is (n - 2) * 180° / n. For example, a hexagon (n = 6) has interior angles of (6 - 2) * 180 / 6 = 120°. The sum of all interior angles is (n - 2) * 180°.

**Q: What happens as the number of sides increases?**

A: As n increases, the regular polygon becomes more circular. Its area approaches π * r², its perimeter approaches 2π * r, and its interior angle approaches 180°. A circle can be thought of as a regular polygon with infinitely many sides.

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Source: https://vastcalc.com/calculators/math/regular-polygon
Category: Math
Last updated: 2026-04-21