Polygon Diagonals and Angles Calculator

The number of diagonals equals the number of sides times the number of sides minus three, all divided by two, written n times n minus 3, over 2. Each of the n corners connects to every other corner except itself and its two neighbours, which is n minus 3 diagonals per corner, and dividing by two removes the double counting.

A diagonal joins two corners that are not already next to each other. In a triangle every pair of corners is joined by a side, so there is nothing left to join. The formula agrees: 3 times 3 minus 3 is 3 times 0, which is 0.

The interior angles add up to the number of sides minus two, times 180 degrees. You can see why by slicing the shape into triangles from one corner: an n-sided polygon splits into n minus 2 triangles, and each triangle carries 180 degrees.

Yes. Each interior angle is the angle sum divided by the number of sides, and each exterior angle is 360 divided by the number of sides. Those single figures only apply when every angle is equal, which is the regular case. An irregular polygon has the same angle sum but unequal individual angles.

Walk once around the boundary of any convex polygon and you turn through a full circle by the time you return to the start. Those turns are the exterior angles, so they total 360 degrees no matter how many sides there are, which is why each one in a regular polygon is 360 over n.

When the number of sides does not divide evenly, the angles are not whole numbers. A heptagon gives 900 over 7 for the interior angle, so the tool rounds to six decimal places and shows 128.571429 degrees rather than a repeating decimal.

A whole number from 3 to 1000. Fractions and decimals are rejected because a polygon cannot have a fractional side, and anything below 3 is refused because you need at least three sides to enclose a region.

No. This is a numeric calculator, not a drawing tool. It returns the diagonal count and the three angle figures as text you can copy, without rendering the shape itself.

It runs entirely in your browser with a couple of arithmetic steps, so it works offline once the page has loaded and the number you enter is never sent anywhere.