Calculate interior angles, angle sums, and exterior angles of regular and custom polygons. Uses the (n−2)×180° formula. Supports polygons from 3 to 100 sides with visual breakdowns.
The interior angles of a polygon are the angles formed inside the shape at each vertex. The sum of all interior angles of an n-sided polygon is given by the elegant formula (n − 2) × 180°. For a triangle (n = 3), the sum is 180°; for a quadrilateral, 360°; for a pentagon, 540°; and so on, increasing by 180° with each additional side.
For regular polygons — where all sides and angles are equal — each interior angle is simply the total sum divided by the number of sides: (n − 2) × 180° / n. A regular hexagon has interior angles of 120°, a regular octagon has 135°, and a regular dodecagon (12 sides) has 150°.
This calculator works in two modes. In regular polygon mode, you enter the number of sides and instantly see each interior angle, the angle sum, the corresponding exterior angle, and how many triangles the polygon can be divided into. In custom polygon mode, you can enter individual angle values for an irregular polygon, and the calculator verifies whether they sum correctly, flags any errors, and computes the missing angle if one is left blank.
The tool includes presets for common polygons from triangle through icosagon (20 sides), a reference table comparing angle properties across polygon types, and visual bars showing the interior/exterior angle ratio at each polygon size. It is an essential resource for geometry students, architects designing tiling patterns, and engineers working with multi-sided shapes.
The Interior Angles Calculator — Regular & Custom Polygons is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Polygon, Each Interior Angle, Sum of Interior Angles in one pass, with conversions and derived values shown together.
Use it to validate homework steps, check CAD or fabrication dimensions, estimate material requirements, and sanity-check hand calculations before submitting work.
Sum of interior angles: S = (n − 2) × 180° Each interior angle (regular): θ = (n − 2) × 180° / n Each exterior angle (regular): 360° / n Number of triangles formed: n − 2 Interior + Exterior = 180° Sum of exterior angles: 360° (always)
Result: Each interior angle = 135°, Angle sum = 1080°, Exterior angle = 45°
A regular octagon has 8 sides. Sum = (8 − 2) × 180° = 6 × 180° = 1080°. Each interior angle = 1080° / 8 = 135°. Each exterior angle = 180° − 135° = 45°. The octagon can be divided into 6 triangles.
This calculator combines the core geometry formula with the input mode selected in the interface, then derives companion values shown in the output cards, comparison bars, and reference tables. Use it to cross-check both direct calculations and reverse-solving scenarios where one measurement is unknown.
Interior Angles Calculator — Regular & Custom Polygons calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.
Keep units consistent across every input before interpreting area, perimeter, angle, or volume outputs. For best results, measure carefully, round only at the final step, and compare at least one manual calculation with the calculator output when building confidence.
The sum of interior angles of an n-sided polygon is (n − 2) × 180°. For example, a hexagon: (6 − 2) × 180° = 720°.
Divide the total sum by the number of sides: (n − 2) × 180° / n. For a regular pentagon: 540° / 5 = 108°.
A regular polygon has all sides equal and all angles equal. An irregular polygon has sides and/or angles of different measures. The angle sum formula works for both.
Each additional side adds one more triangle when the polygon is triangulated. Since each triangle contributes 180°, the sum increases by 180° per side.
Only three: equilateral triangles (60° × 6 = 360°), squares (90° × 4 = 360°), and regular hexagons (120° × 3 = 360°). The interior angle must divide evenly into 360°.
Each interior angle approaches 180°, each exterior angle approaches 0°, and the polygon becomes indistinguishable from a circle. Use this as a practical reminder before finalizing the result.