Polygon Interior/Exterior Angles Calculator — Sum & Per-Vertex Angles
Diagram & Inputs
Regular: enter n (≥3). • Coordinates: click to add points, drag to adjust, or paste x,y pairs below.
Results
Vertices (x, y)
| # | x | y | Actions |
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Paste or Import
Formats: “x,y” or “x y” per line. Scientific notation OK.
How the Polygon Angles Calculator Works
Regular mode: with n sides, each interior angle is \((n-2)\cdot 180^\circ/n\); each exterior angle is \(360^\circ/n\). The interior sum is \((n-2)\cdot 180^\circ\), exterior sum is \(360^\circ\).
Coordinates mode: for vertices ordered around the boundary, the interior angle at vertex \(i\) is computed from adjacent edge vectors. Reflex (concave) angles are correctly identified using the polygon’s orientation (from the signed area). Signed exterior (turning) angles are reported as \(180^\circ-\text{interior}\) and sum to \(\pm 360^\circ\) for simple polygons.
- Interior sum (simple polygon): \(\sum \theta_i = (n-2)\cdot 180^\circ\)
- Exterior sum (signed): \(\sum \varepsilon_i = \pm 360^\circ\) (CCW \(+360^\circ\), CW \(-360^\circ\))
Self-intersection triggers a warning since “interior” can be ambiguous.
Polygon Angles: FAQs
Do I need ordered points?
Yes. Use Auto-order for a quick radial sort if you pasted scattered points; for complex shapes, adjust manually.
Why are some interior angles > 180°?
Those are reflex angles at concave vertices. The tool identifies them using orientation and cross products.
What do signed exterior angles mean?
They’re the turning angles (\(180^\circ - \theta_i\)) with sign from orientation; their sum is \(+360^\circ\) (counterclockwise) or \(-360^\circ\) (clockwise).
Is my data private?
Yes. Everything runs locally; nothing is uploaded.