Interior angles of a polygon: formula, calculator and worked example

What the symbols mean

$S$: the sum of the interior angles, in degrees (°)
$n$: the number of sides (3 or more)

When to use it

Use it to find a missing angle in a polygon, the size of each angle in a regular polygon, or how many sides a regular polygon has from one of its angles. It works for any simple polygon, regular or not, convex or not. The exterior angles of any convex polygon always add up to 360°.

Worked example

What do the interior angles of a hexagon add up to? (n = 6)

Start from the formula.
$S = (n - 2) \cdot 180$
Substitute $n$ = $6$ and work it out.
$S = (6 - 2) \cdot 180$
$= 4 \cdot 180$
$= 720 °$
If the polygon is regular (all angles equal), each interior angle is
$each angle = \frac{(n - 2) \cdot 180}{n}$
$= \frac{(6 - 2) \cdot 180}{6}$
$= \frac{4 \cdot 180}{6}$
$= \frac{720}{6}$
$= 120 °$
and each exterior angle is
$exterior = \frac{360}{n}$
$= \frac{360}{6}$
$= 60 °$

Answer S = 720°; in a regular hexagon each angle is 120°

Why it works

From one corner, draw diagonals to every other corner you can. They split an n-sided polygon into $n − 2$ triangles, and each triangle's angles add up to 180°.

Common mistakes

Using $n×180°$ . Only $n − 2$ triangles fit, not n.
Dividing by n for an irregular polygon. The angles are only equal when the polygon is regular.
Mixing up interior and exterior angles: at each corner they add up to 180°.

Interior angles of a polygon

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