Find an interior-angle sum or each angle of a regular polygon. The displayed angle result includes enough working to inspect signs and scale.
The sum formula checks polygon sketches and supports tile, frame, and pattern layouts. Equal individual angles follow only when the polygon is regular.
Drawing diagonals from one vertex partitions an n-sided polygon into n − 2 triangles. Since each triangle contributes 180°, the interior-angle sum is (n − 2) × 180°.
On this page, the displayed angle result follows this definition rather than any similarly named measure.
An octagon divides into six triangles, so its sum is 6 × 180° = 1080°. A regular octagon has eight equal interior angles, each 1080°/8 = 135°.
Equal sides alone or equal angles alone may not establish regularity in every informal drawing. Do not divide the sum by n unless the problem states that all interior angles are equal. If the Polygon Interior Angle assumptions do not fit, consider regular polygon area.
When polygon interior angle looks surprising, restore the sample and vary number of sides by itself.
Subtract 2 from the whole-number side count and multiply by 180°. For a regular polygon, divide that sum by the number of sides. A hand-worked extension of Polygon Interior Angle is triangle angle sum.
Write each source value under its matching label before calculating: number of sides and result wanted. This preserves the assumptions behind angle result and makes a later check possible without reopening the original problem.
Each added side increases the total interior-angle sum by 180°. For regular polygons, each individual angle approaches—but never reaches—180° as the side count grows. This relationship remains useful even when the final angle result is rounded.
Interior-angle sum grows with side count; consistently oriented exterior angles always total 360°. These are complementary perspectives but not interchangeable outputs. The two results may share inputs while retaining different meanings.
Before publishing or sharing angle result, decide on units, significant digits, and whether an exact form is expected. Those choices belong to interpretation rather than the calculator engine.
When reporting the answer, state the angle result first, then its value and unit. Add number of sides and result wanted if someone else must verify the work independently.
A triangle, with three sides.
No. It applies to simple Euclidean polygons; regularity is only needed to find each equal angle.
Taking one consistently oriented exterior angle at each vertex gives a 360° sum.
Yes for a simple concave polygon, though the visual triangulation is less obvious.
A fan of diagonals from one vertex connects to all nonadjacent vertices, creating two fewer triangles than sides.