🔷Angles in Polygons

A triangle has an interior angle sum locked at 180°. But what about quadrilaterals, pentagons, hundred-gons? No panic — as long as you can cut a shape into triangles, you can reckon every angle it owns.

Built out of triangles

Pick any vertex inside a polygon and draw diagonals to the vertices it cannot reach directly. A quadrilateral splits into 2 triangles, a pentagon into 3, a hexagon into 4 — an $n$-sided polygon always splits into $n-2$ triangles. Each triangle contributes 180°, so the interior angle sum is

$S = (n-2) \times 180^\circ$

Pentagon and hexagon

Pentagon, $n = 5$:

$(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$

Hexagon, $n = 6$:

$(6-2) \times 180^\circ = 720^\circ$

Every extra side adds 180° to the sum — exactly one new triangle's worth.

Regular polygons: every angle alike

In a regular polygon all sides and all angles are equal, so each interior angle is the sum shared out:

$\frac{(n-2) \times 180^\circ}{n}$

Regular pentagon: $540^\circ \div 5 = 108^\circ$. Regular hexagon: $720^\circ \div 6 = 120^\circ$.

Angle60

Type：Acute= 1.047 rad

180° = π rad

Drag the dial to $108^\circ$ — exactly one interior angle of a regular pentagon. Its supplement, $72^\circ$, is one exterior angle of that same pentagon. By the way, every quadrilateral has an interior angle sum of $(4-2) \times 180^\circ = 360^\circ$.

Exterior angles always make 360°

Imagine walking around the edge of a polygon: at each corner you turn once, and after a full lap you face the direction you started — you have turned through 360° in total. Remarkably, this holds no matter how many sides: true for triangles, true for hundred-gons. For a regular polygon each exterior angle is $360^\circ \div n$: the pentagon's $72^\circ$ exterior and $108^\circ$ interior fit together into one straight line. Walking the lap completes the other half of the 180° "half-turn" story.

Check yourself

✅

Quick quiz

↺ Try again

1. 1. What is the interior angle sum of a hexagon?

   540°720°900°

2. 2. What is each interior angle of a regular pentagon?

   60°108°120°

3. 3. No matter how many sides, the exterior angles always sum to

   180°360°a value depending on n