Interior Angles of a Polygon — Formula & Sum

The interior angles of a simple $n$-sided polygon add up to

$$(n-2) \times 180^\circ.$$

If the polygon is regular, each interior angle is

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

That answers the two searches most readers have: the total interior-angle sum for any simple polygon, and one interior angle only when the polygon is regular.

Interior angles are the angles inside the polygon

An interior angle is the angle formed inside a polygon where two sides meet.

For a triangle, the three interior angles add to $180^\circ$. For a quadrilateral, they add to $360^\circ$. The polygon formula extends that same pattern.

Why the polygon interior-angle sum is $(n-2) \times 180^\circ$

A simple way to see the formula is to split the polygon into triangles by drawing diagonals from one vertex.

An $n$-sided polygon can be divided into $n-2$ triangles this way, and each triangle has angle sum $180^\circ$. So the total interior-angle sum is

$$(n-2) \times 180^\circ.$$

This argument works for simple polygons. That includes both convex and concave polygons, as long as the sides do not cross.

Worked example: interior angles of a hexagon

Find the sum of the interior angles of a hexagon.

A hexagon has $n=6$ sides, so

$$(6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ.$$

So the interior angles of any simple hexagon add to $720^\circ$.

If the hexagon is regular, then all six angles are equal, so each one is

$$\frac{720^\circ}{6} = 120^\circ.$$

This is the key distinction:

• Any simple hexagon has interior-angle sum $720^\circ$.
• Only a regular hexagon has every interior angle equal to $120^\circ$.

Common mistakes with polygon interior angles

Dividing by $n$ for a non-regular polygon

The formula

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

gives one interior angle only when the polygon is regular. An irregular pentagon still has total angle sum $540^\circ$, but its individual angles do not have to be equal.

Using the formula without checking whether the polygon is simple

The standard sum formula is for simple polygons. If sides cross, the angle relationships are different, so you should not apply the formula automatically.

Mixing up interior and exterior angles

Interior angles are inside the polygon. Exterior angles are formed outside it. Those are related ideas, but they are not the same quantity.

When the interior-angle formula is used

Interior-angle formulas appear in geometry classes, drawing and design problems, and any situation where you need to reason about polygon shape from side count.

They also matter when you move from a general polygon to a regular polygon, because that is when a total angle sum turns into a single repeated angle.

Try a similar polygon angle problem

Try an octagon with $n=8$.

First find the interior-angle sum using $(n-2) \times 180^\circ$. Then, only if you assume the octagon is regular, divide by $8$ to find one interior angle. If you want the next step, explore another polygon case and check whether you used the regular-polygon condition at the right moment.