Polygon Angle Sums

When working with polygons, understanding how their angles relate to the number of sides is a fundamental geometry skill. Whether you are dealing with a simple triangle or a complex 12-sided figure, specific formulas allow you to find the sums of their interior and exterior angles.

Sum of Interior Angles

The interior angles are the angles inside the closed shape of a polygon. The sum of the interior angles of any $n$-sided polygon is given by the formula:

$\text{Sum} = (n - 2) \times 180^\circ$

Why does this work? If you pick one vertex of a polygon and draw diagonals to all other non-adjacent vertices, you will divide the polygon into exactly $n - 2$ triangles. Since the interior angles of a single triangle always add up to $180^\circ$, multiplying the number of triangles by $180^\circ$ gives the total angle sum.

Example: Find the sum of the interior angles of a 12-gon.

• Identify the number of sides: $n = 12$
• Apply the formula: $S = (12 - 2) \times 180^\circ$
• Calculate: $S = 10 \times 180^\circ = 1800^\circ$

Sum of Exterior Angles

An exterior angle is formed by extending one side of the polygon outward. The rule for exterior angles is beautifully simple:

The sum of the exterior angles of any convex polygon (taking one at each vertex) is always $360^\circ$, regardless of how many sides the polygon has.

Regular Polygons

A regular polygon has all sides equal in length and all interior angles equal in measure. Because the angles are identical, you can easily find the measure of a single angle by dividing the total sum by the number of sides, $n$.

Each Interior Angle: $\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}$

Each Exterior Angle: $\text{Exterior Angle} = \frac{360^\circ}{n}$

(Note: The interior and exterior angle at the same vertex always add up to $180^\circ$.)

Example: Find each interior angle of a regular octagon.

• An octagon has $n = 8$ sides.
• Total interior sum: $(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ$
• Divide by 8: $\frac{1080^\circ}{8} = 135^\circ$

Alternatively, you can find the exterior angle first ($\frac{360^\circ}{8} = 45^\circ$) and subtract it from $180^\circ$ ($180^\circ - 45^\circ = 135^\circ$). Both methods work perfectly!