Interior Angles of a Polygon — Formula & Sum

Need the angle sum of a pentagon, or one angle of a regular hexagon? Both come from the same routine: count the sides, apply $(n-2) \times 180^\circ$, and divide by $n$ only if the shape is regular. For a simple $n$-sided polygon the interior angles always add to

and a regular polygon splits that sum evenly so each interior angle is

When to reach for this method

Use the angle-sum formula whenever you know the side count of a simple polygon (one whose sides do not cross) and need either the total of its interior angles or, in the regular case, a single repeated angle. It covers both convex and concave shapes, and it is the standard tool for geometry, drawing, and design problems that reason about shape from side count alone. If the sides cross, the formula no longer applies directly.

The procedure, step by step

1. Count the sides. Let the number of sides be $n$.
2. Find the angle sum. Apply $(n-2) \times 180^\circ$ to get the total of the interior angles.
3. Check for regularity. Only if the polygon is regular, divide the sum by $n$ to get one interior angle.
4. Verify against a familiar case. Compare with known results such as $180^\circ$ for a triangle or $360^\circ$ for a quadrilateral.

Why step 2 works: drawing diagonals from a single vertex splits an $n$-sided polygon into $n-2$ triangles, each with angle sum $180^\circ$. Multiply and you get $(n-2) \times 180^\circ$.

A full run-through: the hexagon

Take a hexagon, so $n = 6$.

Step 1 fixes $n = 6$. Step 2 gives the sum:

Every simple hexagon has interior angles totaling $720^\circ$. Step 3 applies only if the hexagon is regular; then all six angles match, so each is

The distinction is the whole point:

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

Step 4: $720^\circ$ is consistent with the pattern that grows by $180^\circ$ each time you add a side.

Where each step trips people up

• At "find the angle sum," dividing by $n$ too early. The expression $\frac{(n-2) \times 180^\circ}{n}$ returns one angle only for regular polygons. An irregular pentagon still totals $540^\circ$, but its individual angles need not be equal. Self-check: did the problem actually say "regular"?
• At "count the sides," skipping the simple-polygon condition. If the sides cross, the angle relationships change and the formula should not be applied automatically.
• Throughout, confusing interior with exterior angles. Interior angles sit inside the polygon; exterior angles are formed outside it. Related, but not the same quantity. Self-check: is your angle inside the shape?

FAQ

Use the steps above on an octagon ($n = 8$): get the interior-angle sum with $(n-2) \times 180^\circ$, then divide by $8$ for one angle only if you are told it is regular. The moment you reach for that division is exactly where the regular-polygon condition has to be checked.