Angle Sums and Properties of Polygons

Why the angles in a triangle add to 180°

Draw a triangle and a line through its top vertex parallel to its base. The two angles at the base match angles on the new line because they are alternate angles on parallel lines. Together with the angle at the top of the triangle, they fill a straight line. Angles on a straight line add to $180°$, so the three interior angles of every triangle add to $180°$.

You can use this result to find a missing angle. If two angles in a triangle are $48°$ and $67°$, the third is $180°-48°-67°=65°$.

Building polygons from triangles

A polygon is a closed, flat shape with straight sides. In a convex polygon, draw diagonals from one vertex to all vertices that are not next to it. A pentagon splits into three triangles; a hexagon splits into four. In general, a polygon with $n$ sides splits into $n-2$ triangles, giving an interior-angle sum of

$S=(n-2)\times180°.$

For example, a hexagon has $6-2=4$ triangles, so its interior angles total $720°$. The same angle-sum rule applies to a simple concave polygon, although it may need to be divided into triangles differently.

Interior and exterior angles

Extend one side of a polygon past a vertex. The angle outside, between that extension and the next side, is an exterior angle. It and the interior angle at that vertex lie on a straight line, so they add to $180°$.

For a convex polygon, take one exterior angle at each vertex as you travel around it in the same direction. The exterior angles add to $360°$. You can see why from the interior-angle sum: all $n$ interior–exterior pairs total $n\times180°$, while the interior angles total $(n-2)\times180°$. The difference is $360°$. It is also the total turn made in one complete journey around the polygon.

What changes for a regular polygon?

All the angles of a regular polygon are equal. If it has $n$ sides, divide the interior-angle sum by $n$ to find one interior angle:

$\text{interior angle}=\frac{(n-2)\times180°}{n}.$

Its exterior angles are equal too, so each is $360°\div n$. These two angles must still add to $180°$. For a regular hexagon, each exterior angle is $60°$ and each interior angle is $120°$. Conversely, if each exterior angle of a regular polygon is $45°$, it has $360°\div45°=8$ sides.

Triangle and polygon sums

The interior angles of a triangle total $180°$: a parallel line through one vertex brings all three angles onto a straight line. Splitting an $n$-sided polygon into $n-2$ triangles gives

$\text{interior-angle sum}=(n-2)\times180°.$

Exterior and regular-polygon angles

An interior angle and its adjacent exterior angle total $180°$. One exterior angle at each vertex, taken in the same direction around a convex polygon, totals $360°$.

For a regular $n$-sided polygon:

$\text{one exterior angle}=\frac{360°}{n},\qquad \text{one interior angle}=\frac{(n-2)\times180°}{n}=180°-\frac{360°}{n}.$