Sum of Interior Angles (Polygon)

Statement

The sum of the interior angles of a polygon with \(n\) sides is given by:

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

Here, \(n\) is the number of sides of the polygon, and \(S\) is the total sum of its interior angles.

Why it’s true

• Any polygon can be divided into triangles by drawing diagonals from one vertex.
• A polygon with \(n\) sides can be split into \(n-2\) triangles.
• Since each triangle’s interior angles sum to \(180^\circ\), multiplying by \(n-2\) gives the total sum.

Recipe (how to use it)

1. Identify the number of sides \(n\).
2. Substitute into the formula \(S = (n-2) \times 180^\circ\).
3. Calculate to find the total sum of interior angles.
4. If the polygon is regular, divide \(S\) by \(n\) to find each interior angle.

Spotting it

These problems often mention polygons like pentagon, hexagon, or decagon. Look for “sum of interior angles” or “size of each angle in a regular polygon”.

Common pairings

• Exterior angle rule: each exterior angle of a regular polygon = \(360^\circ / n\).
• Regular polygons: interior angle = \((n-2) \times 180^\circ / n\).

Mini examples

1. Given: Find the sum of interior angles of a hexagon. Answer: \(S = (6-2) \times 180^\circ = 720^\circ\).
2. Given: Find each interior angle of a regular octagon. Answer: \((8-2) \times 180^\circ / 8 = 135^\circ\).

Pitfalls

• Forgetting to subtract 2 before multiplying by 180°.
• Mixing up sum of angles with each individual angle.
• Using exterior instead of interior angle formula.

Exam strategy

• Always check if the polygon is regular (all angles equal) or irregular.
• Write down the formula before substituting values.
• Remember: interior + exterior = 180° for each vertex.

Summary

The sum of interior angles of an \(n\)-sided polygon is \((n-2) \times 180^\circ\). For a regular polygon, divide by \(n\) to find each angle. This formula connects polygon geometry to triangle angle sums.