Polygon Interior Angles CalculatorFree online math calculator — instant results, no sign-up
Find the sum and regular-polygon value of interior angles.
About the Polygon Interior Angles Calculator
The Polygon Interior Angles Calculator finds the sum of a polygon’s interior angles and the size of each interior angle when the polygon is regular. Enter the number of sides to get both results in degrees. With the default of 6 sides, the angle sum is 720° and each regular angle is 120°.
- Number of sides
- The number of sides, n, is a unitless whole-number count of at least 3; the default is 6.
Formula and method
S = (n − 2) × 180°; A = S / n = ((n − 2) × 180°) / n.
- nNumber of sides of the polygon.
- SSum of all interior angles, in degrees.
- AEach interior angle of a regular polygon, in degrees.
A simple polygon can be divided into n − 2 triangles, each with an angle sum of 180°. Multiply n − 2 by 180° to obtain the polygon’s interior-angle sum, then divide that sum by n to obtain each angle for a regular polygon.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Number of sides6
Results
- Angle sum (degrees)720
- Each regular angle (degrees)120
How to use the Polygon Interior Angles Calculator
- 1Enter a whole number of at least 3 in Number of sides; leave the default at 6 to follow the worked example.
- 2Calculate the interior-angle sum using S = (6 − 2) × 180° = 720°.
- 3Divide the sum by the number of sides: A = 720° / 6 = 120°.
- 4Read Angle sum (degrees) as 720 and Each regular angle (degrees) as 120; use the latter as an individual angle only if the polygon has equal interior angles.
Reading your result
Angle sum (degrees) is the total of all interior angles: for 6 sides, it is 720°. Each regular angle (degrees) is the size of every interior angle in a regular polygon: for 6 sides, it is 120°. For an irregular polygon, the sum still applies, but dividing by the number of sides gives only the average interior angle, not necessarily any individual angle.
Limitations
The formulas assume a simple, non-self-intersecting polygon with straight sides in a flat Euclidean plane and no holes. The number of sides must be a whole number of at least 3. The angle sum applies to both convex and concave simple polygons, but the individual-angle output assumes equal interior angles, as in a regular polygon. Side count alone cannot determine the individual angles of an irregular polygon, and these formulas do not cover polygons on curved surfaces.
Common questions
Does a polygon with 6 sides always have interior angles of 120°?+
No. Its interior angles sum to 720° if it is a simple planar polygon, but each angle is 120° only when all its interior angles are equal, as in a regular hexagon.
Can I use the angle sum for a concave polygon?+
Yes. S = (n − 2) × 180° applies to simple concave polygons as well as convex ones. The regular-angle result does not describe the individual angles of a concave polygon.
Why does the angle-sum formula subtract 2 from the number of sides?+
A simple polygon can be triangulated into n − 2 triangles without adding vertices. Adding their 180° angle sums gives the polygon’s total interior-angle sum.
Do I need to enter side lengths to calculate the interior angles?+
No. The number of sides alone determines the total interior-angle sum and each angle of a regular polygon. For an irregular polygon, additional geometric information is needed to determine individual angles.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
You might also like
Slope Calculator
Slope, angle and line equation through two points.
Area Calculator
Area of common 2D shapes.
Volume Calculator
Volume of common 3D shapes.
Surface Area Calculator
Surface area of 3D shapes.
Circle Calculator
Radius, diameter, circumference and area.
Triangle Calculator
Solve a triangle from three sides.
Results are estimates for informational purposes and are not financial, medical or legal advice.