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 |
|---|
Paste or Import
Formats: “x,y” or “x y” per line. Scientific notation OK.
How the Polygon Angles Calculator Works
Want a quick, reliable way to find polygon angles without hunting for formulas? This calculator is built to do exactly that. Whether you are working with a regular polygon like a hexagon, or an irregular shape drawn from coordinates, it computes interior and exterior angles, totals, and turning direction so you can trust the geometry behind your design, homework, or measurement.
The core idea: every polygon angle comes from how much you “turn” as you move from one side to the next. For a regular polygon, that turn is the same at each corner, which gives the familiar formulas for interior angles, exterior angles, and their sums. For an irregular polygon, each corner can differ, so the calculator uses the coordinates of adjacent sides to measure the angle at each vertex and determines whether it is convex or reflex (greater than \(180^\circ\)).
In regular mode, enter the number of sides and the tool returns the interior angle, exterior angle, and the total interior angle sum. In coordinates mode, you provide the vertices in order around the boundary, and the calculator evaluates each corner using vector geometry. It also checks the polygon’s orientation to report signed exterior angles and to identify concave corners correctly.
How to use it:
- Choose regular or coordinates mode based on your data.
- For regular polygons, enter the number of sides.
- For coordinates, add or paste each vertex in order (clockwise or counterclockwise).
- Run the calculation to see interior angles, exterior angles, and totals.
- Review any warnings if the shape self-intersects, since interior angles become ambiguous.
Why it’s useful: understanding polygon angles supports drafting, architecture, tiling patterns, map making, game design, and classroom geometry. You might be checking a CAD sketch, validating a floor plan, or exploring how a pentagon differs from a decagon. The calculator provides fast confirmation and reduces mistakes when dealing with shapes that would be tedious to compute by hand.
For simple polygons, the interior angle sum always follows \((n-2)\cdot 180^\circ\), and the exterior angles add up to a full \(360^\circ\) turn. Those relationships help you sanity-check results, especially when working with many-sided figures.
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.