Polygon Angle
Calculate interior angle, exterior angle, and sum of angles for any regular polygon. Enter the number of sides for instant geometry results with formulas and degrees.
About This Calculator
Use this polygon angle calculator to instantly find the interior angle of any regular polygon from the number of sides. Simply enter the number of sides and get the exact interior angle measurement in degrees.
The calculation uses the standard polygon angle formula: Interior Angle = (n - 2) x 180 deg / n, where n is the number of sides. This formula comes from the fact that any polygon can be divided into (n - 2) triangles, each with 180 deg of interior angles. For example, a regular octagon (8 sides) has interior angles of (8 - 2) x 180 deg / 8 = 135 deg. The sum of all interior angles is (n - 2) x 180 deg.
Regional Notes: Angle measurements in degrees are universal and do not differ by region. For radian conversions, multiply degrees by pi/180. This calculator supports any polygon from a triangle (3 sides) to very large numbers of sides (approaching a circle at 180 deg).
Frequently Asked Questions
How do you calculate the interior angle of a regular polygon?
Each interior angle of a regular polygon is (n - 2) x 180 degrees / n, where n is the number of sides. For a regular pentagon (5 sides): (5 - 2) x 180 degrees / 5 = 3 x 180 degrees / 5 = 108 degrees. For a square (4 sides): (4 - 2) x 180 degrees / 4 = 90 degrees.
What is the sum of interior angles of a polygon?
The sum of the interior angles of any polygon is (n - 2) x 180 degrees, where n is the number of sides. For a triangle (n = 3): 180 degrees. For a quadrilateral (n = 4): 360 degrees. For a pentagon (n = 5): 540 degrees. This formula works for both regular and irregular polygons.
What is the interior angle of a regular hexagon?
A regular hexagon has 6 sides. Using the formula (n - 2) x 180 degrees / n = (6 - 2) x 180 degrees / 6 = 4 x 30 degrees = 120 degrees. So each interior angle of a regular hexagon is 120 degrees.
What is the difference between interior and exterior angles?
The interior angle is the angle inside the polygon at each vertex. The exterior angle is the supplementary angle formed by extending one side outward. For any regular polygon, interior + exterior = 180 degrees. The sum of all exterior angles of any polygon is always 360 degrees. The exterior angle = 360 degrees / n.
How many sides does a polygon have if each interior angle is 156 degrees?
Using the formula (n - 2) x 180 degrees / n = interior angle. Set (n - 2) x 180 / n = 156. Solve: 180n - 360 = 156n, so 24n = 360, n = 15. So a regular polygon with 156-degree interior angles has 15 sides -- a regular pentadecagon.
How do you find the exterior angle of a regular polygon?
The exterior angle of a regular polygon is 360 degrees divided by the number of sides (n). For a square (4 sides): 360/4 = 90 degrees. For a regular pentagon (5 sides): 360/5 = 72 degrees. For a regular hexagon (6 sides): 360/6 = 60 degrees. The sum of all exterior angles of any polygon is always 360 degrees, regardless of the number of sides.
Is the Polygon Angle Calculator free to use?
Yes, all calculators on Calculy are completely free to use. No registration, login, or payment is required. You can bookmark any calculation with shareable URLs that preserve your inputs.