16.4 Angles in polygons — Learn

Prerequisites: use triangle and quadrilateral angle sums and collect like terms.

Objectives: calculate the interior sum of an $n$-sided polygon, its regular interior angle, and the exterior angle sum. This covers G3.

Interior angle sums

A polygon is a closed shape made from straight sides. An $n$-sided polygon can be split from one vertex into $n-2$ triangles. Therefore its interior angle sum is $(n-2) imes180^\circ$.

Example 1. Find the interior angle sum of a hexagon.

A regular polygon has equal sides and equal angles. Divide its angle sum by its number of angles to find one interior angle.

Example 2. Find each interior angle of a regular hexagon.

Exterior angles

At each vertex, an interior angle and its exterior angle make a straight line. Taking one exterior angle at each vertex gives a total of $360^\circ$.

Example 3. Find each exterior angle of a regular octagon.

Watch out. The interior angle sum is not $360^\circ$ for every polygon. A quadrilateral has $360^\circ$, but a pentagon has $(5-2) imes180=540^\circ$. Check the number of triangles, $n-2$.

Progress check

A regular courtyard tile has six sides. Predict each interior angle, then use the formula to check it.

You should now be able to

• use $(n-2) imes180^\circ$ for an interior angle sum;
• find one interior angle of a regular polygon;
• use the $360^\circ$ exterior angle sum for regular or irregular polygons.