Interior and Exterior Angle Sum Calculator for Polygons

Math Geometry • Basic Shapes and Properties

Written by STEM Calculators Team Published January 21, 2026 Updated February 24, 2026

Find the sum of interior angles and sum of exterior angles for any polygon. For a regular polygon, you can also compute the measure of each interior/exterior angle.

Inputs accept pi, e, sqrt(2), 1e-3. Use * for multiplication. (Angles are reported in degrees by default.)

Enter \(n\) and click Solve.

Polygon angle visualization (pan/zoom enabled)

Pan: drag empty space • Zoom: wheel/trackpad • “Reset view” restores default.

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Interior and Exterior Angle Sums of Polygons

For a polygon with \(n\) sides (where \(n \ge 3\)), two classic results are: the sum of interior angles and the sum of exterior angles.

Sum of interior angles

Any simple polygon can be triangulated into \(n-2\) triangles. Each triangle has angle sum \(180^\circ\), so:

\[ S_{\text{int}} = (n-2)\cdot 180^\circ. \]

In radians, this is \(S_{\text{int}} = (n-2)\pi\).

Sum of exterior angles

If you walk around the boundary of a simple polygon (no self-intersections), you make exactly one full turn, so the total exterior turning is:

\[ S_{\text{ext}} = 360^\circ \quad (\text{or } 2\pi \text{ radians}). \]

This is true even for irregular polygons, as long as the boundary is a single non-self-intersecting loop.

Regular polygons (each angle)

For a regular polygon, all angles are equal:

\[ \alpha = \frac{S_{\text{int}}}{n} = \frac{(n-2)\cdot 180^\circ}{n}, \qquad \beta = \frac{S_{\text{ext}}}{n} = \frac{360^\circ}{n}, \] where \(\alpha\) is each interior angle and \(\beta\) is each exterior angle. In radians: \[ \alpha = \frac{(n-2)\pi}{n}, \qquad \beta = \frac{2\pi}{n}. \]

Visualization features

Interior wedge at a vertex (visualizes an interior angle).
Exterior turning arrow (visualizes the turning angle at a vertex).
Walk-around animation: illustrates that the total exterior turning equals \(360^\circ\) (or \(2\pi\)).
Triangulation overlay (optional): draws diagonals from one vertex to show the \(n-2\) triangles used in the interior-sum derivation.

Notes

Convex vs. concave: the sums above still hold for simple concave polygons.
Self-intersecting polygons: the exterior sum can differ depending on winding number (not covered here).
Units: degrees are often used in geometry; radians are common in calculus/physics.

Frequently Asked Questions

What is the formula for the sum of interior angles of a polygon?

For an n-sided polygon, the interior-angle sum is S_int = (n - 2) x 180° in degrees. In radians it is S_int = (n - 2) x pi.

Why is the sum of exterior angles always 360 degrees?

Walking once around the boundary of any simple (non-self-intersecting) polygon makes exactly one full turn. One full turn is 360° (or 2 x pi radians), so the exterior-angle sum is constant.

How do you find each interior and exterior angle of a regular polygon?

In a regular polygon all angles are equal, so each interior angle is (n - 2) x 180° / n and each exterior angle is 360° / n. The same relationships hold in radians using pi.

When should I use degrees vs radians for polygon angles?

Degrees are common in geometry problems, while radians are standard in calculus and many physics formulas. The calculator can report the same results in either unit.

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Frequently Asked Questions

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