Polygon Angle Calculator - Interior, Exterior, Sums

A polygon angle calculator is a geometry tool that returns the interior angle, exterior angle, total angle sums, polygon name, diagonal count, and triangle count of any regular polygon from a single side-count input. Pick a number of sides between 3 and 1000 and the calculator applies the standard (n-2)*180 identities to give you a complete angle summary of the shape in degrees.

• • Homework and contest geometry: Solve problems that ask for the interior or exterior angle of a regular pentagon, hexagon, or dodecagon without redoing the same arithmetic by hand.
• • Tessellation and tiling checks: Check whether a regular polygon tiles the plane (interior angle divides 360) before committing to a tile pattern.
• • Construction and joinery: Pick a regular polygon for a frame, fence, or coffered ceiling and read the miter angle for each joint directly from the side count.
• • 3D modeling and CAD baselines: Use the angle values to set up regular prisms, pyramids, and other shapes in CAD without rounding errors from a calculator app.

The two identities that drive the tool are the interior angle of a regular n-gon, (n-2)*180 / n degrees, and the exterior angle, 360 / n degrees. The two add up to 180 because the interior and exterior angle at a vertex of a convex polygon are supplementary, and the n exterior angles always sum to 360.

Use this polygon angle calculator whenever a regular polygon is defined by its side count and you need a fast angle, diagonal, or triangle-fan count. For a triangle, the triangle calculator gives a more direct route to angles from side lengths.

Once you have the angle values for a regular polygon, the regular Polygon Area Calculator returns the area, perimeter, and apothem from a single side length.

The calculator reads the n input, rounds it to the nearest integer, and runs the regular-polygon identities in a single pass to produce every angle and count in the result panel.

• n: Number of sides of the regular polygon. Must be a positive integer of at least 3.
• (n-2)*180: Sum of the interior angles, which is the basis for both per-vertex angle formulas.
• (n-2)*180 / n: Each interior angle of the regular polygon, found by dividing the sum evenly across all n vertices.
• 360 / n: Each exterior angle, which is also 180 minus the interior angle since the two share a straight line at each vertex.
• 360: Sum of the exterior angles, which is a constant for any convex polygon regardless of the side count.

The exterior-angle sum of 360 degrees is what makes the exterior-angle identity so useful: you do not need to know anything about the polygon other than n to recover the per-vertex exterior angle. Dividing 360 by n gives the angle between a side and the extension of the adjacent side, which is also the miter angle for a regular frame joint.

The diagonal count identity, n(n-3)/2, drops out of the same combinatorial picture. Each of the n vertices can connect to n-3 non-adjacent vertices, and every diagonal is counted twice from both ends, so the count is n(n-3)/2. A triangle has 0 diagonals, a quadrilateral has 2, and the count climbs quickly as n grows.

According to Wikipedia, the sum of the interior angles of a simple n-gon is (n-2)*180 degrees, and each interior angle of a regular n-gon is (n-2)*180/n.

For the most common regular polygon, the Hexagon Calculator returns the area, perimeter, apothem, and side length of a regular hexagon from a single length input.

Four ideas hold the angle formulas together. Knowing them makes the result easier to interpret for unusual side counts.

The relationship between the interior sum (n-2)*180 and the number of triangles n-2 is the geometric reason the formula works. Cutting the polygon from one vertex into triangles turns the interior-angle sum into the sum of n-2 straight triangles, which is (n-2)*180 degrees. The same identity is what the calculator reports as the trianglesFromOneVertex output.

When the polygon is not regular, Irregular Polygon Area Calculator takes an ordered list of vertex coordinates and returns the area, perimeter, and centroid using the shoelace formula.

Pick a side count, watch the result panel update, and read the angle, sum, diagonal, and triangle values for the regular polygon.

1. 1 Enter the number of sides: Type the side count n in the input field. The minimum is 3 (a triangle); larger values work the same way up to 1000.
2. 2 Read the polygon name: The result panel shows the common name for the side count (triangle, quadrilateral, pentagon, ... up to icosagon, then n-gon).
3. 3 Read the per-vertex angles: The interior and exterior angles of the regular n-gon are reported in degrees. They add up to 180 at every vertex.
4. 4 Check the sums and counts: Use the interior sum (n-2)*180, the constant exterior sum of 360, the diagonal count n(n-3)/2, and the triangle fan count n-2 for your work.

For a regular dodecagon (n=12), the calculator returns polygon name = dodecagon, interior = 150 deg, exterior = 30 deg, interior sum = 1800 deg, exterior sum = 360 deg, diagonals = 54, and triangles from one vertex = 10. That is enough information to set up a regular 12-sided frame with 150-degree joints or to check that twelve 30-degree exterior angles close to a full turn.

When the side count is exactly 3, Triangle Calculator is the right next step: it returns the third angle and the side lengths from any two pieces of triangle data.

A single integer input gives you a full description of a regular polygon's angles and combinatorial structure, which is faster than redoing the same arithmetic for every new shape.

• • Covers any regular polygon: Triangles through 1000-gons are handled by the same formula, with the polygon name resolved automatically for the most common shapes.
• • Returns a complete angle summary: Interior angle, exterior angle, interior sum, and exterior sum are all in the result panel so there is no need to chain separate calculations.
• • Includes diagonal and triangle counts: The n(n-3)/2 diagonal count and the n-2 triangle-fan count come out alongside the angles, which is useful for tiling and graph problems.
• • Works for very large n: The formula is exact for any integer n. Enter 360 and you read the interior angle of a regular 360-gon without any rounding.
• • Free of generic app noise: There is no signup, no clutter, and no unit toggle to worry about. The output is always in degrees, which matches how angle problems are written.

For students the main win is that one input replaces three or four separate computations. For builders, designers, and CAD users the win is that the miter and bevel angles for a regular frame are available the moment the side count is set.

When the regular polygon happens to be a decagon, switching to Decagon Area Calculator gives the area, perimeter, apothem, and circumradius from one side length without typing 10 vertices.

The angle formulas are exact, but a few conditions decide what the calculator is allowed to report. These are the most common inputs to watch out for.

• • n must be a positive integer of at least 3. n < 3 returns a validation error rather than a fabricated answer.
• • The per-vertex interior and exterior angles are exact only for regular polygons. For an irregular polygon, the same sum still holds, but each individual angle is generally different from the (n-2)*180 / n value.

If your shape is a triangle, the formula reduces to 60 degrees per interior angle. The triangle calculator is a more direct route from sides and angles for that smallest case. For all other regular polygons, this calculator is the fastest way to the same answer.

According to Wolfram MathWorld, a regular n-gon has n vertices, n edges, and n(n-3)/2 diagonals, with each interior angle equal to (n-2)*pi/n radians.

When the same angle needs to be expressed in radians, gradians, or turns, Angle Converter converts between degrees, radians, and other common angle units in one step.