Polygon Angle Calculator

Enter the number of sides of a regular polygon to find its interior angle sum, each interior angle, and each exterior angle using (n − 2) × 180° and 360° / n.

Quick Facts

Sum of interior angles
(n − 2) × 180°
Found by splitting an n-sided polygon into (n − 2) triangles from one vertex.
Each interior angle (regular)
(n − 2) × 180° / n
Only valid when every side and angle is equal.
Each exterior angle (regular)
360° / n
Interior and exterior angles at a vertex are supplementary (sum to 180°).
Sum of exterior angles
360°, always
True for any convex polygon regardless of the number of sides.

Your Results

Calculated
Sum of Interior Angles
-
(n − 2) × 180°
Each Interior Angle
-
Regular polygon: sum ÷ n
Each Exterior Angle
-
Regular polygon: 360° ÷ n
Sum of Exterior Angles
-
Always 360° for a convex polygon

Ready

Enter a number of sides and pick a unit, then press Calculate.

Formula and Method for Polygon Interior and Exterior Angles

A polygon's interior angles are the angles formed inside the shape at each vertex, and their total always depends only on the number of sides, n. For any simple polygon with n sides, the sum of the interior angles is (n − 2) × 180°. For a regular polygon (all sides and angles equal), dividing that sum by n gives the measure of each interior angle, and the exterior angle at each vertex — the supplement of the interior angle — equals 360° / n.

How the calculation works

Enter the number of sides, n (at least 3), and choose degrees or radians. The calculator first finds the interior angle sum using (n − 2) × 180°, which comes from splitting any polygon into (n − 2) non-overlapping triangles by drawing diagonals from a single vertex — since every triangle's angles sum to 180°, an n-sided polygon's interior angles sum to (n − 2) × 180°. Assuming the polygon is regular, each interior angle is that sum divided by n, and each exterior angle is 360° divided by n. The exterior angles, one per vertex measured consistently around the perimeter, always sum to a full turn — 360° — no matter how many sides the polygon has.

Common mistakes

  • Assuming irregular polygons have equal angles: the (n − 2) × 180° sum applies to any simple polygon, but dividing by n to get "each" angle is only valid for a regular polygon where every angle is identical.
  • Confusing interior and exterior angles: at each vertex, the interior and exterior angles are supplementary — they add to 180°, not 360°.
  • Applying the rule to concave shapes: the exterior-angle-sum-to-360° rule assumes a convex polygon; concave (non-convex) polygons need signed angles to keep the rule true.

Real-world applications

  • Architects and carpenters use interior angle formulas to cut miters for regular-polygon window frames, gazebos, and tile patterns.
  • CAD and CNC software uses the exterior angle (360°/n) to generate regular polygon paths, bolt patterns, and gears.
  • Surveyors and civil engineers check that a closed traverse's measured interior angles sum to (n − 2) × 180° as an error-detection step.
  • Game and graphics programmers use the exterior angle turn to procedurally generate and rotate regular polygon vertices.

Frequently Asked Questions

What is the formula for the sum of interior angles of a polygon?
The sum of the interior angles of any simple polygon with n sides is (n − 2) × 180°. This comes from dividing the polygon into (n − 2) triangles by drawing diagonals from one vertex, since each triangle contributes 180°. A hexagon (n = 6), for example, has interior angles summing to (6 − 2) × 180° = 720°.
How do I find one interior angle of a regular polygon?
For a regular polygon, divide the interior angle sum by the number of sides: (n − 2) × 180° / n. A regular hexagon has each interior angle equal to 720° / 6 = 120°.
What is the exterior angle of a regular polygon?
Each exterior angle of a regular polygon equals 360° / n, since the interior and exterior angles at a vertex are supplementary and together the exterior angles sweep one full turn around the shape. A regular hexagon's exterior angle is 360° / 6 = 60°.
Why do a polygon's exterior angles always add up to 360°?
Walking once around the perimeter of any convex polygon and turning by the exterior angle at each vertex brings you back to your starting direction after exactly one full rotation — 360° — regardless of how many sides the polygon has.

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