Polygon Angle Calculator

Understanding Polygon Angles: Interior, Exterior, and Sums

A polygon is a closed two-dimensional figure bounded by straight line segments. Whether studying basic geometry or designing architectural structures, computing the angles of polygons is a fundamental mathematical skill. This Polygon Angle Calculator enables you to calculate the interior angle, exterior angle, sum of interior angles, central angle, and number of diagonals for any regular polygon, as well as determine the missing interior angle of irregular polygons.

Interior Angles of a Regular Polygon

An interior angle is an angle formed inside a polygon between two adjacent sides. In any regular polygon with \(n\) equal sides, all interior angles are congruent.

The sum of the interior angles \(S_{\text{int}}\) depends exclusively on the number of sides \(n\) and is given by dividing the polygon into \(n - 2\) non-overlapping triangles:

$$S_{\text{int}} = (n - 2) \times 180^\circ$$

To find the measure of each individual interior angle \(\alpha\) in a regular polygon, divide the total sum by the number of vertices \(n\):

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

For instance, a regular hexagon has \(n = 6\) sides. Its interior angle sum is \((6 - 2) \times 180^\circ = 720^\circ\). Each interior angle measures \(720^\circ / 6 = 120^\circ\). You can compare these properties with our Regular Polygon Calculator.

Exterior Angles and the Constant Sum

An exterior angle is formed between one side of a polygon and the extension of an adjacent side. At each vertex, the interior angle and its corresponding exterior angle form a straight linear pair, which means they are supplementary:

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

A remarkable property of all convex polygons (regardless of how many sides they possess) is that the sum of exterior angles is always constant:

$$S_{\text{ext}} = 360^\circ$$

Consequently, in a regular polygon where all exterior angles are equal, the measure of each exterior angle \(\beta\) is:

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

For specialized triangle calculations, explore our Triangle Calculator and Exterior Angles of a Triangle Calculator.

Diagonals of a Polygon

A diagonal is a line segment connecting two non-consecutive vertices of a polygon. From any given vertex in an \(n\)-sided polygon, diagonals can be drawn to all vertices except itself and the two adjacent neighbors, allowing \(n - 3\) diagonals per vertex. Because each diagonal connects two vertices, the total number of unique diagonals \(d\) is:

$$d = \frac{n(n - 3)}{2}$$

Polygon Angles Quick Reference Table

Finding Missing Angles in Irregular Polygons

In an irregular polygon, sides and interior angles vary in length and measure. However, the total interior angle sum rule still strictly holds:

$$\text{Missing Angle} = (n - 2) \times 180^\circ - \sum(\text{Known Angles})$$

If you have a quadrilateral with three known angles measuring 80°, 110°, and 95°, you subtract their sum (285°) from 360° to find the missing fourth angle: \(360^\circ - 285^\circ = 75^\circ\).