Exterior Angle of Polygon Calculator – Instant Results with Step-by-Step Formulas
An exterior angle of a polygon is the angle formed at a vertex when one side of the polygon is extended beyond that vertex. It is always supplementary to the interior angle at the same vertex, meaning the two angles add up to exactly 180°. This calculator lets you compute exterior angles in four different modes: finding the exterior angle of a regular polygon from its number of sides, converting between interior and exterior angles, and working backwards from an exterior angle to derive the number of sides of a regular polygon.
The Core Formula for Regular Polygons
For any regular polygon with n sides — meaning all sides are equal in length and all angles are equal in measure — the exterior angle at every vertex is:
Exterior Angle = 360° ÷ n
For example, a regular hexagon has 6 sides, so each exterior angle equals 360° ÷ 6 = 60°. A regular pentagon gives 360° ÷ 5 = 72°, and a square (n = 4) produces 360° ÷ 4 = 90°. Because each exterior angle is the supplement of the interior angle, you can also derive the interior angle immediately: Interior Angle = 180° − Exterior Angle.
Why the Sum of Exterior Angles Is Always 360°
One of the most elegant theorems in plane geometry is that the sum of all exterior angles of any convex polygon is always 360°, regardless of the number of sides. The intuitive explanation is to imagine walking around the perimeter of the polygon. At each vertex you turn through the exterior angle. By the time you complete one full circuit and face the same direction as when you started, you have rotated through exactly one full revolution — 360°. This result holds for a triangle, a decagon, or any polygon with any number of sides.
For a regular polygon this means: n × (360° ÷ n) = 360°, which is trivially true. For anirregular convex polygon the individual exterior angles are different at each vertex, but their sum is still 360°. This property makes exterior angles particularly useful in navigation and robotics, where turning angles between path segments must sum to a complete rotation.
Interior to Exterior Conversion
If you know one interior angle of any polygon vertex, you can find the exterior angle at that vertex using the supplementary-angle relationship:
Exterior Angle = 180° − Interior Angle
This works for both regular and irregular polygons. For instance, if one vertex of a polygon has an interior angle of 135°, the exterior angle at that vertex is 180° − 135° = 45°. This mode is particularly helpful in geometry proofs and when verifying the consistency of polygon angle data.
Reverse Calculation: Finding the Number of Sides
Because Exterior Angle = 360° ÷ n, you can rearrange to find the number of sides when you know the exterior angle of a regular polygon:
n = 360° ÷ Exterior Angle
For this to produce a valid regular polygon, the result must be a whole number greater than or equal to 3. If you enter an exterior angle of 40°, you get n = 360 ÷ 40 = 9, which is a valid nonagon. An exterior angle of 45° gives n = 8 (octagon). An exterior angle of 50° gives n = 7.2, which is not an integer, so no regular polygon has exterior angles of exactly 50°.
Practical Applications
Exterior angles appear in several real-world contexts:
- Navigation and robotics: When a robot or vehicle follows a polygonal path, the turning angle at each waypoint equals the exterior angle. The fact that these angles always sum to 360° confirms a closed-loop path.
- Tessellations: Understanding exterior angles helps determine which regular polygons can tile a flat surface. Only polygons whose interior angles are divisors of 360° can tile by themselves: equilateral triangles (60°), squares (90°), and regular hexagons (120°).
- CAD and graphic design: When constructing regular polygons programmatically, the exterior angle tells you the rotation increment between consecutive vertices.
- Geometry proofs: The exterior angle theorem for triangles states that an exterior angle equals the sum of the two non-adjacent interior angles — a result commonly used in proofs and problem-solving.
Using the Calculator
Select one of the four modes at the top of the tool. In Regular Polygon mode, use the preset buttons (Triangle, Quadrilateral, Pentagon, etc.) or type any number of sides from 3 to 1000. The calculator instantly shows the exterior angle, interior angle, sum of interior angles, and a visual SVG diagram highlighting the exterior angle arc at one vertex.
In Interior → Exterior and Exterior → Interior conversion modes, simply enter one angle and get the other. In Sides from Exterior mode, enter the exterior angle and the tool tells you how many sides the corresponding regular polygon has — or explains why no regular polygon matches if the result is not an integer.
You can switch the output between degrees and radians, adjust the displayed decimal precision from 0 to 10 places, copy all results to the clipboard, and export a reference table to CSV for use in spreadsheets or reports.