Free Polygon Angle Calculator

The Polygon Angle Calculator is a free online Interior Angle Calculator and Exterior Angle Calculator designed for regular polygons. Enter the number of sides, choose your preferred unit (degrees or radians), and the tool instantly returns both the interior and exterior angles. It also reinforces the underlying Polygon Angle Formula and the sum of interior angles.

Interior and Exterior Angles Defined

In a regular polygon (a shape where all sides are equal and all angles are equal), interior angles are the angles formed inside the figure at each vertex. Exterior angles are created by extending one side of the polygon and measuring the angle between that extension and the adjacent side; for a regular polygon, every exterior angle has the same measure and equals the central angle.

Formulas for Regular Polygon Angles

For a regular polygon with $n$ sides, the interior angle $\alpha$ is given by:

$$\alpha = \frac{(n-2) \times 180^\circ}{n}$$

In radians, the same expression is:

$$\alpha = \frac{(n-2) \times \pi}{n}$$

The numerator $(n-2) \times 180^\circ$ represents the sum of interior angles for any convex polygon with $n$ sides.

The exterior angle $\beta$ is:

$$\beta = \frac{360^\circ}{n}$$

or in radians:

$$\beta = \frac{2\pi}{n}$$

Because the interior and exterior angles at a vertex lie on a straight line, they are supplementary:

$$\alpha + \beta = 180^\circ$$

If you know one angle, the other can be found by simple subtraction.

Angle Values for Common Regular Polygons

The table below lists interior and exterior angles for popular regular polygons. As the number of sides increases, the interior angle approaches $180^\circ$ while the exterior angle approaches $0^\circ$.

Polygon	Sides $(n)$	Interior Angle	Exterior Angle
Triangle	3	60°	120°
Square	4	90°	90°
Pentagon	5	108°	72°
Hexagon	6	120°	60°
Heptagon	7	≈128.57°	≈51.43°
Octagon	8	135°	45°
Nonagon	9	140°	40°
Decagon	10	144°	36°

Going Backwards: Find the Number of Sides from an Angle

If you are given the exterior angle of a regular polygon, you can determine its number of sides:

$$n = \frac{360^\circ}{\beta}$$

For example, an exterior angle of $72^\circ$ corresponds to $n = 360/72 = 5$, a pentagon. Alternatively, if the interior angle $\alpha$ is known, you can use:

$$n = \frac{360^\circ}{180^\circ - \alpha}$$

Using the Polygon Angle Calculator

To use the tool, simply select degrees or radians, type the number of sides—or supply an angle if you are working backwards—and the calculator does the rest. It is especially useful for students verifying homework, designers checking specifications, or anyone needing quick access to geometry angle data.

An Important Property: Exterior Angles Always Sum to 360°

Regardless of how many sides a convex polygon has, the sum of its exterior angles is always $360^\circ$ (or $2\pi$ radians). This can serve as a consistency check: the product $n \times \beta$ must equal $360^\circ$ for a correct calculation.

Whether you are exploring Regular Polygon Angles for the first time or need a reliable Polygon Angle Formula reference, this Interior Angle Calculator and Exterior Angle Calculator provides all the information you need in seconds.