Exterior angles never change
Walk around any simple polygon and turn the same way each corner: those signed exterior angles always sum to ±360°—no matter how many sides.
Constant turn
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.
5 Fun Facts about Polygon Angles
Triangle is the “angle atom”
The interior sum formula \((n-2)\cdot180°\) comes from chopping any polygon into n−2 triangles. Triangles are the angle building blocks.
Sum origin
Tiling needs tidy angles
To tile the plane with one regular polygon, its interior angle must divide \(360°\). Only 3, 4, or 6 sides qualify—why hexagons get the spotlight.
Pattern rule
Reflex flips the math
A concave vertex has interior > 180°; the turning (exterior) angle goes negative there. One reflex corner can swing the orientation total.
Concave twist
Almost a circle
In a regular polygon, every interior angle creeps toward 180° as sides grow. At 1,000 sides you’re within 0.18° of a perfect circle.
Limit intuition
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.