LifeTally
Regular polygon angles and diagonals
What are the angles and diagonal count of your convex regular polygon with n sides?
Enter the whole side count to find the interior angle, exterior turn, exact interior-angle sum and number of diagonals.
Your polygon side count
The declared shape is flat, simple, convex and regular: straight equal sides and equal interior angles. n is also the number of vertices. The exterior angle here is the turn supplementary to the interior angle, not the reflex angle outside the shape.
How the estimate works
A fan from one vertex has n − 2 triangles, giving S = 180 × (n − 2) degrees. Equal angles give a = S ÷ n. Equal exterior turns share one full turn: b = 360 ÷ n, with a + b = 180 degrees. Of all unordered vertex pairs, n are sides; the rest are diagonals, so D = n × (n − 3) ÷ 2. For n = 3, there are genuinely zero diagonals.
S = 180 × (n − 2)
a = S ÷ n; b = 360 ÷ n
D = n × (n − 3) ÷ 2The side count declares a mathematical shape; it does not check a real drawing. This model excludes irregular, concave, star or self-intersecting polygons, curved sides, spherical and three-dimensional geometry. It does not find side or diagonal lengths, perimeter, area, diagonal intersection points or distinct diagonal lengths. It does not provide cutting, bevel, tiling, installation or safety instructions, or guarantee physical accuracy. No diagram is generated.
OpenStax: regular polygons and angles ↗OpenStax: unordered combinations ↗
Check the filled example
Chosen example: n = 9. Seven triangles give S = 1260 degrees. Each interior angle is 140 degrees and each exterior turn is 40 degrees. There are 36 unordered vertex pairs; removing the nine sides leaves 27 diagonals. This is our checkable example, not a construction specification.
Accepted input values
- Number of sides n
- 9
Plan the next step
Use the standard calculator for separate arithmetic with your own numbers. A numerical result does not verify that a drawing is regular.
Open standard calculator →