The Angle Sum of a Polygon

Look down at a tiled floor, a football stitched from patches, or the honeycomb a bee builds. Corners meet corners, and somehow it all fits together with no gaps and no overlaps. That only works because the angles inside each shape are not random — they add up to a fixed total that you can predict before you ever pick up a protractor.

And the totals follow a beautifully tidy pattern. A triangle's three angles add up to 180^\circ. A quadrilateral's four angles add up to 360^\circ. A pentagon's five add up to 540^\circ. Spot it? Every extra side adds another 180^\circ — one more triangle's worth, every single time.

ShapeSides nAngle sum
Triangle3180^\circ
Quadrilateral4360^\circ
Pentagon5540^\circ
Hexagon6720^\circ
Decagon101440^\circ

You already know that the three angles of a triangle add up to 180^\circ. That single fact unlocks every polygon: a four-sided shape, a five-sided shape, a hundred-sided shape. The trick is to cut the polygon into triangles and count.

Pick one corner of an n-sided polygon and draw a diagonal to each of the other corners. The polygon falls apart into exactly n - 2 triangles, and the polygon's interior angles are just the triangles' angles gathered up:

\text{interior angle sum} = (n - 2) \times 180^\circ For any polygon with n sides:

Why it works

Watch a pentagon (five sides) come apart. From a single corner, two diagonals carve it into three triangles — and 5 - 2 = 3, exactly as the rule predicts.

Every interior angle of the pentagon is built from angles of those triangles, with nothing left over and nothing double-counted. Three triangles, each worth 180^\circ, give 3 \times 180^\circ = 540^\circ. The same cut works for any polygon: an n-gon always splits into n - 2 triangles.

Why n - 2 and not n? Because the corner you start from can't draw a diagonal to itself or to the two corners right next to it (those are already joined by sides). So n corners give n - 3 diagonals, and n - 3 diagonals slice the shape into n - 2 triangles.

The pattern is a straight line

Plot the angle sum against the number of sides and the dots march up a perfectly straight staircase — every step to the right (one more side) is a step of 180^\circ upward. That constant step is the fingerprint of the (n-2)\times 180^\circ rule.

Worked examples

1. The angle sum of a decagon. A decagon has n = 10 sides. Straight into the rule:

(10 - 2) \times 180^\circ = 8 \times 180^\circ = 1440^\circ.

2. A missing interior angle. A pentagon (n = 5, so the angles total 540^\circ) has four angles measuring 100^\circ, 110^\circ, 120^\circ and 95^\circ. Add the known four, then take them from the total:

100 + 110 + 120 + 95 = 425, \qquad 540^\circ - 425^\circ = 115^\circ.

The fifth angle is 115^\circ.

3. Working backward from the total. A polygon's interior angles add up to 900^\circ. How many sides? Set the rule equal to 900 and unwind it:

(n - 2) \times 180 = 900 \;\Rightarrow\; n - 2 = \tfrac{900}{180} = 5 \;\Rightarrow\; n = 7.

A seven-sided polygon — a heptagon.

The formula gives the total of all the interior angles added together — not the size of a single angle. This trips almost everyone up at least once.

If someone asks for each angle of a regular polygon (one where every angle is equal), you must take that total and share it out between the corners — divide by n:

\text{each angle of a regular } n\text{-gon} = \frac{(n - 2)\times 180^\circ}{n}.

A regular hexagon: the total is 720^\circ, so each angle is 720 \div 6 = 120^\circ — not 720^\circ! Mixing up "the whole angle sum" with "one angle" is the classic slip. You'll meet this share-it-out step properly with regular polygons.

Cutting a shape into triangles has a grand name — triangulation — and it is one of the most useful ideas in all of maths. Every 3D character in a video game, every dinosaur in a film, every curved car body in design software is secretly a mesh of thousands of tiny triangles, because a computer finds triangles the easiest shape of all to draw and shade.

The same idea pins down positions. Surveyors mapping a country, and the satellites behind GPS, work out where you are by forming triangles between known points and measuring their angles. Triangles are rigid — they can't wobble out of shape the way a square can — which makes them the trustworthy atoms that every other polygon (and every map) is built from.

See it explained