Interior sum — every simple polygon
S = (n − 2) × 180°
Dividing a polygon from one vertex creates n − 2 triangles.
Polygon Angle Calculator – Interior & Exterior Angles
Find each interior, exterior, or central angle; calculate the interior-angle sum; solve for the number of sides; or find one missing angle in an irregular polygon. Choose regular, irregular, or advanced coordinate mode. Calculations run locally in your browser.
What do you know?
Enter a whole number of at least 3.
Enter exactly n − 1 angles, separated by commas or spaces.
Advanced coordinate analysis. Enter vertices in boundary order, clockwise or counterclockwise. Clicking the diagram can add a point; the table provides a complete keyboard-accessible alternative.
A regular hexagon divided into 4 triangles. α is an interior angle and β is the exterior or central angle.
Answer
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Coordinate vertices
Vertices must follow the polygon boundary in clockwise or counterclockwise order. Do not repeat the first point at the end.
| # | x | y | Actions |
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Use the labelled move and delete buttons to edit the point order without dragging.
Advanced coordinate actions
Centroid-angle sorting is only a starting point for scattered convex point sets. It can create the wrong boundary for concave shapes; inspect the diagram and intersection report afterward.
Polygon angle formulas
Every simple polygon: the interior sum and one-missing-angle formulas apply to regular or irregular simple polygons. Regular polygons only: the per-angle and reverse formulas require all angles to be equal.
Each interior angle — regular only
i = S ÷ n
Equivalent to i = 180° − 360° ÷ n.
Each exterior or central angle — regular only
e = 360° ÷ n
At one vertex, i + e = 180°.
Sides from exterior angle — regular only
n = 360° ÷ e
A valid polygon result must be a whole number of at least 3.
Sides from interior angle — regular only
n = 360° ÷ (180° − i)
The regular interior angle must be at least 60° and less than 180°.
One missing interior angle — every simple polygon
missing = S − Σ known angles
Provide n − 1 known interior angles.
Worked examples
Interior angles of a pentagon
S = (5 − 2) × 180° = 540°. If regular, each angle is 540° ÷ 5 = 108°.
Interior and exterior angles of a hexagon
S = (6 − 2) × 180° = 720°. Each interior angle is 720° ÷ 6 = 120°; each exterior angle is 360° ÷ 6 = 60°.
Polygon with a 60° exterior angle
n = 360° ÷ 60° = 6, so it is a regular hexagon.
Sides from a 150° interior angle
n = 360° ÷ (180° − 150°) = 360° ÷ 30° = 12, a regular dodecagon.
Missing angle in an irregular pentagon
S = 540°. Known sum = 100° + 110° + 120° + 95° = 425°. Missing = 540° − 425° = 115°.
Common regular polygons
| Sides | Name | Interior sum | Each interior | Each exterior |
|---|---|---|---|---|
| 3 | Triangle | 180° | 60° | 120° |
| 4 | Quadrilateral (square) | 360° | 90° | 90° |
| 5 | Pentagon | 540° | 108° | 72° |
| 6 | Hexagon | 720° | 120° | 60° |
| 7 | Heptagon | 900° | 128.571° | 51.429° |
| 8 | Octagon | 1,080° | 135° | 45° |
| 9 | Nonagon | 1,260° | 140° | 40° |
| 10 | Decagon | 1,440° | 144° | 36° |
| 11 | Undecagon | 1,620° | 147.273° | 32.727° |
| 12 | Dodecagon | 1,800° | 150° | 30° |
How advanced coordinate analysis works
The coordinate mode expects a simple boundary path, with each vertex listed once in clockwise or counterclockwise order. It uses the shoelace signed area to determine orientation, dot and cross products to measure each interior angle and reflex status, and segment tests to report exact intersecting edge pairs.
Diagnostics report duplicate vertices, zero-length edges, collinear triples, degenerate area, and self-intersections. The measured interior and signed exterior totals are compared with the theoretical totals. Centroid-angle sorting is deliberately labelled as a tentative aid because it is not reliable for every concave point set.
Review and accuracy
Reviewed by
Starlight Tools Mathematics Editorial Team
Geometry formula and numerical-method review.
Last reviewed: 17 July 2026
Calculation methodology
Regular results use closed-form Euclidean polygon formulas. Irregular missing angles subtract the known sum from (n − 2) × 180°. Coordinate angles use normalized vector dot products, cross-product reflex tests, and orientation-aware signed turns.
Accuracy note
Automated in-page checks cover clockwise, counterclockwise, concave, duplicate-point, collinear, degenerate, and self-intersecting cases. Results are calculated at full floating-point precision, then rounded only for display. Coordinate results can be unreliable for nearly coincident points or invalid boundary order.
References: OpenStax, Polygons, Perimeter, and Circumference and Chapter 10 Formula Review.
Polygon angle FAQs
What is the interior-angle-sum formula?
For every simple n-sided polygon, S = (n − 2) × 180°. A pentagon has S = (5 − 2) × 180° = 540°.
How do I find each angle of a regular polygon?
Divide the interior sum by n: i = (n − 2) × 180° ÷ n. The exterior or central angle is e = 360° ÷ n. For a hexagon, i = 120° and e = 60°.
How do I find the number of sides from an interior angle?
For a regular polygon, use n = 360° ÷ (180° − i). If i = 150°, n = 360° ÷ 30° = 12 sides.
How do I find the number of sides from an exterior angle?
For a regular polygon, use n = 360° ÷ e. If e = 60°, n = 360° ÷ 60° = 6 sides, so the polygon is a hexagon.
How do I find one missing angle in an irregular polygon?
Find S = (n − 2) × 180°, add the known interior angles, then subtract: missing angle = S − known-angle sum. For a pentagon with known angles 100°, 110°, 120°, and 95°, the missing angle is 115°.
Do exterior angles always total 360°?
One exterior turning angle at each vertex of a simple polygon totals one full turn: +360° when traversed counterclockwise or −360° when traversed clockwise. Unsigned regular exterior angles total 360°.
What changes for concave or self-intersecting polygons?
A simple concave polygon still has interior sum (n − 2) × 180°, but reflex interior angles exceed 180° and their signed turns reverse sign. A self-intersecting path is not a simple polygon, so this sum rule may not apply; the coordinate tool reports the intersecting edge pairs.