In this activity, we will explore the interior angles of any polygon with n sides. By the end of the lesson, we will be able to find the sum of the interior angles of any polygon without measuring a single angle![br][br]We learned before that the sum of the interior angles of a triangle is 180 degrees. We will use this throughout our exploration.[br][br]To begin, use the slider to change the number of sides in the polygon below. The label below the polygon will update with the name of the polygon and the sum of the interior angles in the polygon.
Notice what happens to the angle sum as the number of sides increases. What do you think the interior angle sum of a regular nonagon (a regular polygon with 9 sides) will be?
You likely saw a pattern in how the interior angle sum increases as the number of sides increases. Why does this happen? Let's explore the applet again, but with the addition of dividing the polygon into triangles:
Remember that each triangle has an interior angle sum of 180 degrees. If we add up the interior angles of each triangle, the result will be greater than the interior angle sum of the polygon. Where is the extra coming from? How much extra is there?
The interior angle sum for a regular polygon with n sides is [math]180\cdot\left(n-2\right)[/math] degrees. Using what you've observed, how could we arrive at this formula?
So far, we've only looked at regular polygons, where all of the side lengths and angles are equal. What about other polygons? Experiment by dragging the vertices of the hexagon below.
Does our formula still work when the hexagon is irregular? Why or why not?