Students will use their knowledge that the angles of a triangle add to \(180^\circ\) to understand why the interior angles of a simple polygon add up to \((n - 2)180^\circ\). Students will learn both how to derive this formula and how to use it. Here's an excellent derivation of the formula. Students will also consider the special case of regular polygons, and solve problems involving them. One type of problem is to find the number of sides given one interior angle. Another is to find one interior angle given the number of sides. Here's another interesting type of problem:
Next, students will learn what a regular tessellation is, and why there are exactly three regular tessellations. The proof involves nothing more than finding the interior angle of a regular polygon, and knowing that angles around a point add to \(360^\circ.\)
Conclude by giving your students these challenges:
In the figure below, all the blue angles are equal. Mentally determine the angle.
Here's the answer:
Because the 3 pentagons fit together to form a full angle at the center, we know the top angle is \(360^\circ/3 = 120^\circ.\) We also know two of the angles are \(90^\circ,\) so adding those we get \(180^\circ.\) Adding \(120^\circ\) and \(180^\circ\) gives us \(300^\circ,\) so there are \(240^\circ\) degrees left to account for. But the blue angles are equal, so the missing angle must be half that, which is \(120^\circ.\)
Find the green angle:
Here's the answer:
Each interior angle of a heptagon is 128 4/7 degrees. Thus, the blue angle is 128 4/7 - 90 = 38 4/7 degrees. Thus, the orange angle is 90 - 38 4/7 = 51 3/7 degrees. Thus, the red angle is 128 4/7 - 51 3/7 = 77 1/7 degrees. Each interior angle of a pentagon is 108 degrees. Thus, the green angle is 108 - 77 1/7 = 30 6/7 degreees.
There are several ways to solve this challenge, which immediately come to mind.
Mentally determine the green angle:
Here's the answer:
The interior angles of a regular hexagon are \(120^\circ,\) and the interior angles of a square are \(90^\circ.\) We know all the angles surrounding the missing angle, plus the angle itself, sum to \(360^\circ.\) Thus, if we remove the known angles, we'll be left with unknown angle. \(120^\circ \cdot 2 = 240^\circ\) and \(240^\circ + 90^\circ = 330^\circ.\) Thus, our missing angle is \(360^\circ - 330^\circ = 30^\circ.\)