From the Tutor's Corner

Determining the number of degrees in the angles of other polygons can be calculated by dividing the polygon into triangles. A quadrangle (four sides) can be divided into two triangles so that the total of its angles is \(360^{\circ}\). The pentagon below can be divided into three triangles giving \(540^{\circ}\) total degrees, etc.

It is interesting to see what happens when regular polygons (all angles equal) are divided into triangles and the number of degrees in each angle is calculated:

Number of Sides Degrees in Each Angle
\(3\) (triangle) \(60^{\circ}\)
\(4\) (square) \(90^{\circ}\)
\(5\) (pentagon) \(108^{\circ}\)
\(6\) (hexagon) \(120^{\circ}\)
\(7\) (heptagon) \(129^{\circ}\)
\(8\) (octagon) \(135^{\circ}\)
\(\infty\) (infinite} \(180^{\circ}\)

The table shows that as the number of sides of a regular polygon is increased to infinity, the polygon angle approaches \(180^{\circ}\), or a straight line! This implies that a circle can be thought of as a polygon with an infinite number of infinitely short straight sides, a concept often used in calculus.

The above table can also be used to determine which regular polygons can be used to make a tessellate. Since the sum of the angles around a vertex point of a tessellation must be \(360^{\circ}\), only those regular polygons whose angle can be divided evenly into \(360^{\circ}\) will tessellate. As the table shows, the triangle, square, and hexagon are the only regular polygons that can be made into a tessellate.