Does every pentagon have 108° interior corners?

Every simple pentagon has interior sum 540°. Equal 108° corners require the shape to be equiangular; a regular pentagon is one such case. A general pentagon can have unequal corners.

Why can a rounded corner fail the inverse test?

Dividing by 180° minus the angle can amplify rounding, especially near a straight angle. The strict tolerance avoids claiming that an approximate angle describes an exact integer side count.

Can a concave polygon have an interior angle above 180°?

A concave simple polygon has at least one reflex interior corner and still follows the same total-sum formula. The one-corner inverse mode is specifically for convex regular polygons, so it excludes those reflex values.

Interior Angle Calculator

The interior angle calculator answers two related polygon questions: how large is the sum of the inside corner angles, and how large is each corner when the polygon is regular? Enter a whole number of sides to see both results, or enter one regular interior angle to test whether it corresponds to a valid whole side count. These questions require different assumptions, so the output identifies the regular-corner value explicitly.

Separate a total from a corner with the interior angle calculator

An interior angle lies inside a polygon where neighboring sides meet. A simple polygon has a closed boundary that does not cross itself. Its interior-angle sum depends only on its number of sides. Individual corners need not be equal. A regular polygon has equal sides and equal interior angles, which makes dividing the sum by the side count appropriate. Equal-angle polygons also share that angle value, even when their side lengths differ.

Calculate from sides or test a known corner

Select Number of sides and enter an integer from 3 through 10,000. The count refers to boundary segments, not diagonals, triangles inside the shape, or marked points along a straight edge.
Read Interior angle sum as the total for a simple polygon. Read One regular interior angle as the equal corner size under regularity. For an irregular shape, the latter is only the average of the corner sizes.
Select Regular interior angle to infer a side count. Enter degrees from 60° inclusive to 180° exclusive. A finite convex regular polygon has no corner reaching a straight angle.
Check inverse validation before accepting an answer. The tool does not turn a fractional number of sides into a valid polygon by rounding. For repeating angle values, use the most accurate available input rather than a short decimal copied from a display.

Interior-angle formulas and the triangle argument

A simple n-sided polygon can be partitioned into n − 2 triangles. Each triangle contributes 180°, giving sum S = (n − 2) × 180°. For a convex polygon, diagonals from one vertex illustrate the partition directly. Concave simple polygons can also be triangulated, but those diagonals need not all come from one chosen vertex.
For a regular polygon, each interior angle I = S / n = 180° − 360° / n. As n increases the corners approach 180° from below. No finite side count produces exactly 180°.
Rearranging gives n = 360 / (180 − I). The inferred number must be an integer in the supported range. The existing site's inverse helper accepts an estimate only within 0.0000001 of an integer, and checks the reconstructed angle within 0.0000001 degree.
The sum formula applies to simple polygons, including concave ones. It does not directly apply to self-intersecting star boundaries, polygons with holes treated as one region, or curved surfaces with non-Euclidean geometry. The regular corner calculation assumes a planar convex polygon.

Corner and total examples

A five-sided polygon has sum (5 − 2) × 180° = 540°. If it is regular, each corner is 540° / 5 = 108°. An irregular pentagon still has total 540°, but a particular corner cannot be determined from five sides alone.
A regular octagon has sum 1,080° and each interior angle 135°. In inverse mode, n = 360 / (180 − 135) = 8, so the angle produces an exact whole side count.
A given 100° regular interior angle implies n = 360 / 80 = 4.5. The input lies inside the broad angular domain but does not describe an exact regular polygon. The tool reports that distinction instead of suggesting four or five sides.
A regular heptagon has a repeating corner value of approximately 128.571429°. A rounded six-place value can fail the deliberately strict inverse tolerance. Use its fuller value when testing the inverse, or enter seven sides directly to calculate the angle without that ambiguity.

When the drawing is conceptual

The SVG draws actual regular vertices through 24 sides and highlights one interior corner. Beyond that, a labeled conceptual circle keeps the diagram compact and the page responsive. The equations still use the entered side count; the simplified image does not imply that a polygon is mathematically a circle.

Students can use the total to check missing-corner exercises, while designers can compare ideal regular outlines. To find a particular unknown corner in an irregular polygon, subtract all other known interior angles from the total. An average alone cannot identify that missing corner.

Rounding, regularity, and missing information

The interior angle calculator uses unrounded calculations and displays ordinary decimals to at most six places. Extreme magnitudes use six significant digits. An approximate displayed angle does not supply more measurement precision than its input, and a count inferred within tolerance is a numerical consistency check, not proof that a drawn or physical polygon is regular. Fractional counts, fewer than three sides, and counts above 10,000 are rejected. Use the polygon hub for names, diagonals, and central-angle comparisons. Use the exterior page for the turning angle along an extended side; equal numerical relationships do not place the angles in the same geometric region.