Areas of Regular Polygons
Construction
| Step 1: | Construct a Circle (circle with Center through Point) |
| Step 2: | Construct Segment AB. The Radius of circle A. We are going to inscribe a regular pentagon into our circle. If the polygon is inscribed in a circle, it means that the vertices of the polygon lie on the circle. Since we are constructing a regular pentagon, how many points do we need on the circle? ________ |
| Step 3: | Now we will use our rotation tool (Rotate around Point) to create points on the circle. How many degrees are in a circle? _____. In order to have 5 points on the circle, we need to divide the amount of degrees in a circle by the amount of points we want on the circle and this will give us the angle of rotation. What is the angle of rotation for a regular pentagon? _____. |
| Step 4: | Rotate around Point : select the segment AB then the Center A, then rotate the segment AB by the angle of rotation you found in Step 3. Continue rotating until you return to the point you started with. You should now have five points on your circle, along with give congruent radii. |
| Step 5: | Use your segment tool to connect the points around the circle to form your regular pentagon. |
Investigation
How many triangles did you form in your construction?
It turns out that all of these triangles are congruent. (We will prove this later.) What is the formula in terms of b and h for the area of one of these triangles?
The apothem of a regular polygon is the perpendicular segment from the center of the polygon to a side of the polygon. What does it mean for a segment to be perpendicular to another segment? _____________________________. This segment is also the altitude of each of the congruent triangles that form the polygon. Write a formula for the area of one of the triangles in terms of the apothem a and thelength of a side s.
Use the formula from question 3 to write a formula for the area of the entire pentagon in terms of the apothem, a, and the length of the side, s.
What is the perimeter of this regular polygon in terms of a side, s?
Look back at your formulas for question 4 and 5. Do you notice any common terms? What is the common term in each equation?
Now let p be the common term from question 6. Rewrite theequation in question 4 in terms of the perimeter p and theapothem a. You just discovered a theorem!