Sizing Up the Area of a Polygon
Not only can polygons be classified by the number of sides they have and by their angles, but they can also be grouped according to some of their qualities. Polygons can have three personality characteristics: equilateral, equiangular, and regular.
In an equilateral polygon, all sides are equal and there's at least one nonsimilar angle. In an equiangular polygon, all angles are equal and at least one side doesn't match the length of the others. A regular polygon is both equilateral and equiangular; it has total symmetry — equal sides and equal angles.
Some lines are special
When you set out to find the area of a regular polygon, you've got to keep in mind that regular polygons have lines with special meaning. These lines include the radius and the apothem.
The radius is a line that goes from the center of the polygon into an elbow (or vertex if you prefer the technical babble) of the polygon — splitting that angle evenly into two. When two different radii in a polygon are drawn to two consecutive vertices, a central angle is formed in the center of the polygon (see Figure 1).
Unlike the radius, which intersects an angle, an apothem runs from the center of the polygon straight into a flat side of the polygon. On impact, the apothem becomes a perpendicular bisector of the side it collides with (see Figure 2).
A feast of theorems
A slew of theorems exist for radii, central angles, and apothems of regular polygons. Here's a summary for your reading pleasure:
- Theorem 5-8: Radii of a regular polygon bisect the interior angle.
- Theorem 5-9: Central angles of a regular polygon are congruent.
- Theorem 5-10: Central angles of regular polygons with equal sides are congruent.
- Theorem 5-11: The measure of a central angle in a regular polygon is equal to 360° divided by the number of sides of the polygon.
- Theorem 5-12: An apothem of a regular polygon bisects the central angle (determined by the side) to which it's drawn.
- Theorem 5-13: An apothem of a regular polygon is a perpendicular bisector to the side it's drawn to.
Putting it all together
You can calculate the area of a regular polygon by using the length of its apothem and the length of its perimeter: You need to survey the perimeter and determine its length. Use the information about the length of one side. Because the polygon is regular, the lengths are the same for each side. Multiply the number of sides of the polygon by the length of one side, and you get the perimeter. The area of a regular polygon is equal to one-half the product of the apothem and the perimeter.
Theorem 5-14: The formula for the area of a regular polygon is A = 1/2ap, where a is the apothem and p is the perimeter.
Translation: If you have a regular polygon, plug the length of the apothem and the perimeter into the formula, and you get the area.
Take a look at Figure 3 for an example. The information given indicates that the length of one side of the pentagon equals 5 and that the apothem equals 6. Before you can determine the area, you must first calculate the perimeter. If the length of one side of a pentagon is 5, then the perimeter is equal to a side length of 5 multiplied by five sides. So the total perimeter of the pentagon equals 25. If you plug this information into the area formula, you get the following:
A = 1/2 (6)(25)
A = 1/2 (150)
A = 75
So the area of the pentagon in Figure 3, with the given information, is 75 square units.
Now consider this: Just as you can add line segments and angles, you can also add areas.
Postulate 5-1: If a polygon encloses smaller, non-overlapping regions within its perimeter, then the area of that polygon is equal to the sum of the areas of the enclosed regions.
Take a look at the concave polygon in Figure 4. To find the total area of the figure, get the area of sections that you can easily obtain. Look closely: You can actually break the polygon into two non-overlapping rectangles. Find the area of each rectangle and then add them together. You then have the area of the whole polygon.