# Polygon Angle Calculator

Calculate interior and exterior angles of any regular polygon. Find angle sums and area from number of sides. Free online polygon angle calculator.

## What this calculates

Calculate the interior angle, exterior angle, and angle sum of any regular polygon. Optionally enter the side length to compute the area. Works for any polygon from a triangle (3 sides) to a 1000-gon.

## Inputs

- **Number of Sides** — min 3, max 1000 — The number of sides (minimum 3 for a triangle).
- **Side Length (optional)** — min 0 — Optional: enter the side length to calculate the area of the regular polygon.

## Outputs

- **Interior Angle** — Each interior angle of the regular polygon.
- **Exterior Angle** — Each exterior angle of the regular polygon.
- **Sum of Interior Angles** — The total of all interior angles.
- **Area** — The area of the regular polygon (requires side length).

## Details

A regular polygon has all sides equal and all interior angles equal. The formulas for angles depend only on the number of sides (n):

Polygon Angle Formulas

- Sum of Interior Angles: (n - 2) x 180 degrees

- Each Interior Angle: (n - 2) x 180 / n degrees

- Each Exterior Angle: 360 / n degrees

- Area (with side s): (n x s^2) / (4 x tan(pi/n))

The exterior angles of any convex polygon always sum to exactly 360 degrees, regardless of the number of sides. As the number of sides increases, the polygon approaches a circle, and each interior angle approaches 180 degrees.

Common regular polygons include the equilateral triangle (3 sides, 60 degrees interior), square (4 sides, 90 degrees), pentagon (5 sides, 108 degrees), hexagon (6 sides, 120 degrees), octagon (8 sides, 135 degrees), and dodecagon (12 sides, 150 degrees).

## Frequently Asked Questions

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

A: Use the formula: Interior Angle = (n - 2) x 180 / n, where n is the number of sides. For a regular pentagon (n=5): (5-2) x 180 / 5 = 3 x 180 / 5 = 108 degrees. Each interior angle is 108 degrees.

**Q: Why do exterior angles always sum to 360 degrees?**

A: Imagine walking along the perimeter of a convex polygon. At each vertex, you turn through the exterior angle. After completing the full circuit, you face your original direction, having made exactly one full rotation (360 degrees). This is true regardless of the number of sides.

**Q: What is the sum of interior angles for common polygons?**

A: Triangle: 180 degrees, Quadrilateral: 360 degrees, Pentagon: 540 degrees, Hexagon: 720 degrees, Heptagon: 900 degrees, Octagon: 1080 degrees. The formula is (n-2) x 180, adding 180 degrees for each additional side.

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

A: As n increases, the regular polygon approaches a circle. The interior angle approaches 180 degrees, the exterior angle approaches 0 degrees, and the area (for a fixed side length) grows. A 100-sided polygon has interior angles of 176.4 degrees and looks nearly circular.

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