Sum of Interior Angles Calculator – Any Polygon
The sum of interior angles of a polygon is the total measure of all angles formed inside the shape at its vertices. This value depends entirely on the number of sides n — not on the polygon's size, proportions, or whether its sides are equal. Whether you are working with a perfectly regular hexagon or an irregular twelve-sided tile layout, the total always follows the same rule.
The Core Formula
For any convex polygon with n sides, the sum of all interior angles is:
Sum = (n − 2) × 180°
In radians the equivalent is Sum = (n − 2) × π. The formula comes directly from triangulation: any convex polygon can be cut into exactly n − 2 triangles by drawing non-crossing diagonals from a single vertex. Each triangle contributes exactly 180°, so the total is always (n − 2) × 180.
Quick Reference by Polygon Type
- Triangle (n = 3): (3 − 2) × 180 = 180°. Every triangle in the plane, no matter how irregular, has angles summing to exactly 180°.
- Quadrilateral (n = 4): (4 − 2) × 180 = 360°. Squares, rectangles, parallelograms, trapezoids — all share this total.
- Pentagon (n = 5): 540°. Each angle of a regular pentagon is 108°, which allows them to tile surfaces in combination with other shapes.
- Hexagon (n = 6): 720°. Each angle of a regular hexagon is 120°. Three hexagons meet at every vertex (3 × 120 = 360°), which is why hexagonal tiles pack perfectly without gaps.
- Octagon (n = 8): 1080°. Interior angles of regular stop-sign octagons are 135° each.
- Decagon (n = 10): 1440°. Each angle of a regular decagon is 144°.
- Dodecagon (n = 12): 1800°. Used in floor tile patterns and decorative art; each regular angle is 150°.
Each Interior Angle of a Regular Polygon
When a polygon is regular (all sides and all angles equal), every interior angle has the same measure. Divide the total sum by the number of sides:
Each angle = (n − 2) × 180 / n
As n grows, this value approaches 180° from below — the polygon increasingly resembles a circle. A regular 100-gon has interior angles of 176.4°, while a regular 1000-gon has angles of 179.64°.
Verifying an Irregular Polygon
The sum formula holds for any simple convex polygon, regular or irregular. If you have a polygon with known angles — for instance from a survey drawing, a hand-drawn design, or a set of measurements — you can confirm correctness by summing all angles and comparing to (n − 2) × 180. This tool's Irregular mode accepts each angle individually and flags any discrepancy immediately.
A common use case is checking architectural floor plans: if a room outline has five walls (a pentagon), the sum of all five interior angles must equal 540°. A discrepancy signals a measurement or drafting error.
Degrees and Radians
Geometry education typically works in degrees, while advanced mathematics, physics, and most programming environments default to radians. The conversion between the two is:
radians = degrees × π / 180
A hexagon's interior angle sum of 720° equals 4π ≈ 12.5664 rad. This calculator lets you switch between both units at any time, making it equally useful for classroom geometry and computational geometry work.
Why the Sum Grows Linearly
Each time you add one more side to a polygon, the sum of interior angles increases by exactly 180°. This linear relationship — one extra triangle per extra side — is why a comparison table for polygons produces a perfectly spaced sequence: 180, 360, 540, 720, …, with a constant step of 180.
Practical Applications
Architecture and Interior Design
Designers building polygonal rooms, skylights, or decorative panels need the angle sum to compute miter cuts and joint angles. A regular octagonal bay window requires eight corners at 135° each — verifiable instantly with this calculator.
Tile and Mosaic Work
Tiles tessellate (tile a flat surface without gaps) only when the interior angles at each meeting point sum to exactly 360°. Knowing the interior angle of each polygon shape allows artisans to plan which combinations will fit together seamlessly.
CAD and Computational Geometry
Software generating or validating polygonal meshes often checks that angle sums per face equal the expected value. Deviations indicate degenerate or non-planar geometry. Developers can use this tool to quickly look up expected values for testing.
Education
Students learning polygon properties can use the comparison table to observe the linear growth pattern, verify homework answers, and build intuition for why triangles, quadrilaterals, and higher polygons behave differently. The step-by-step formula breakdown shows exactly where the (n − 2) × 180 figure comes from.
Limits and Boundary Cases
- Minimum n = 3 (triangle): angles sum to 180°. No closed polygon exists with fewer sides.
- Concave polygons: the formula
(n − 2) × 180still gives the correct sum for simple (non-self-intersecting) concave polygons, even though some individual angles exceed 180°. Self-intersecting polygons are a different case. - As n → ∞: the polygon approaches a circle, and each interior angle approaches 180°, while the total sum grows without bound.