6) Angles of a Polygon

There are many kinds of polygons ... each with angles and sides. When we talk about the angles of a polygon, we usually mean the interior angles. A polygon has as many interior angles as sides. An equilateral triangle has three equal 60 degree angles. The sum of the angles of this and any triangle is 180 degrees.The sum of the four interior angles of a square is 360 degrees, which is the same for any quadrilateral. The sum of the interior angles increases by 180 degrees for each additional side ... pentagon, hexagon, heptagon, octagon ... as the polygons have more sides, the interior angles become larger and there are more of them, so the sum of the interior angles increases.

There are other angles related to polygons: the exterior angles. An exterior angle is formed by extending one side at each vertex. As the number of sides increases, the exterior angles become smaller. What do you suppose will happen to the sum of the exterior angles? Take the hexagon for example. Look at the exterior angles, and particularly their sum. If we make smaller and smaller green hexagons in the center, we see that the sum of the exterior angles is ...a full circle, 360 degrees. This is true for any polygon .

To find the measure of each exterior angle of a regular polygon, you just divide 360 degrees by the number of sides. So the measure of each exterior angle of an equilateral triangle is 120 degrees, the measure of each exterior angle of a regular quadrilateral (a square) is 90 degrees, and the measure of each exterior angle of a regular pentagon is 72 degrees.

Therefore we can use this information to find the measure of each interior angle of a regular polygon! If we look at the relationship between one interior angle of a polygon and the exterior angle at that vertex, we see that together they make a straight angle, 180 degrees. Therefore, if we know the measure of an interior angle, we can find the measure of its adjacent exterior angle by subtracting from 180 degrees. For example, in an equilateral triangle, the exterior angle is 360 divided by 3 angles = 120 degrees. So each interior angle is 180 - 120 = 60 degrees.

Test question #5: What is the measure of each exterior angle, and each interior angle, of a regular hexagon?

We see these polygons in ornamental grill and tile patterns, and some polygons have special properties that make them more usable for this than others.The square is so perfectly symmetrical that squares (and rectangles) fit together perfectly. This is why so many buildings and other structures are made of squares and rectangles.


Go to Topic 7 Tessellations