In this target, we will focus on finding areas of regular polygons. Regular polygons are special, in that they can be inscribed in a circle. A polygon is inscribed in a circle if you can construct a circle that goes through each of the vertices of the polygon. You can also say that the circle is circumscribed about the polygon. See the picture below.
Once the polygon is inscribed in the circle, we have a few more vocabulary words we can now discuss. A central angle is an angle whose vertex is the center of the circle and whose edges go through the vertices of the polygon (see below). To find the measure of each central angle, you can divide \(360^{\circ}\) by the number of sides, \(n\), in the polygon:
Measure of central angle = \(\dfrac{360^{\circ}}{n}\)
The apothem is segment that goes from the center of the circle to the midpoint of a side of the polygon. The apothem is perpendicular to the side (see below). The radius of the regular polygon is a segment whose endpoints are the center of the circle and a vertex of the polygon (see below).
Measure of central angle = \(\dfrac{360^{\circ}}{n}\)
The apothem is segment that goes from the center of the circle to the midpoint of a side of the polygon. The apothem is perpendicular to the side (see below). The radius of the regular polygon is a segment whose endpoints are the center of the circle and a vertex of the polygon (see below).
Please note: Though only one central angle is shown, there are a total of \(5\) central angles (each with the same measure) that can be drawn in the pentagon above. The same works for the apothem and radius.
Area of Regular Pentagon
There are a few ways to calculate the area of a polygon. You discovered a couple of formulas to calculate the area of a triangle in the Investigations.
Area of Regular Pentagon
There are a few ways to calculate the area of a polygon. You discovered a couple of formulas to calculate the area of a triangle in the Investigations.
Area of a Regular Polygon
\(\dfrac{\text{(apothem length)(perimeter of the polygon)}}{2}\)=\(\dfrac{aP}{2}\)
or
\(\text{(Area of central triangle)(number of sides of the polygon)}\) = \((A_{central \triangle})(n)\)
\(\dfrac{\text{(apothem length)(perimeter of the polygon)}}{2}\)=\(\dfrac{aP}{2}\)
or
\(\text{(Area of central triangle)(number of sides of the polygon)}\) = \((A_{central \triangle})(n)\)
Example 1:
Given a regular heptagon whose radius is \(6.9\) cm and perimeter is \(42\) cm, calculate the area. Solution: Since the heptagon is regular, we know that all of the sides are congruent. With the perimeter\(42\) cm, we can divide \(42\) by \(7\) to determine the length of each side, \(6\) cm. |
To use the above equation, we will need to find the length of the apothem. When we sketch the apothem, recall that it bisects the side length of the regular polygon.
Now that we have the length of the apothem, we can substitute that length into the area formula:
\(A = \dfrac{aP}{2}\)
\(A = \dfrac{(6.214)(42)}{2}\)
\(A = 130.494\) cm\(^2\)
or
\((A_{central \triangle})(n)\)
\(A = \left(\dfrac{(6)(6.214)}{2}\right)(7)\)
\(A = 130.494\) cm\(^{2}\)
\(A = \dfrac{aP}{2}\)
\(A = \dfrac{(6.214)(42)}{2}\)
\(A = 130.494\) cm\(^2\)
or
\((A_{central \triangle})(n)\)
\(A = \left(\dfrac{(6)(6.214)}{2}\right)(7)\)
\(A = 130.494\) cm\(^{2}\)
Sometimes when finding the area of a polygon, we don’t have enough information to use the Pythagorean theorem. In this case, we will useangles to help us find the area of the polygon.
Example 2: Find the area of a regular hexagon whose perimeter is \(48\) feet. Solution: Since the hexagon is regular, we can find the length of each side by dividing by the number of sides in a hexagon. The length of each side is \(8\) feet. If we draw in the apothem, the side length is split in half. But now notice we do not have another side length to use the Pythagorean Theorem. We will find some angles in order to give us more information. There are a few ways to find angle measures, so we will show you each way: |
Option 1: Find the measure of an interior angle
Step 1: Each interior angle = \(\dfrac{(180(n-2))}{n}\) =\(\dfrac{180(6-2)}{6}\) = \(120^{\circ}\) Step 2: Split the hexagon into \(6\) congruent isosceles triangles. Notice this bisects the interior angle:
Step 3: Find the area of one triangle or the length of the apothem. Notice that the apothem splits the triangle into \(30^{\circ}-60^{\circ}-90^{\circ}\) triangles. Use your \(30-60-90\) patterns to find the length of the apothem. Don’t remember the patterns? Go back to look at them here! Step 4: Calculate the area of the polygon: |
Option 2: Find the measure of an exterior angle.
Step 1:
Each exterior angle = \(\dfrac{360^{\circ}}{6}=60^{\circ}\)
Step 1:
Each exterior angle = \(\dfrac{360^{\circ}}{6}=60^{\circ}\)
Step 2:
Find the measure of the interior angle, which is supplementary to the exterior angle = \(180-60=120^{\circ}\)
Find the measure of the interior angle, which is supplementary to the exterior angle = \(180-60=120^{\circ}\)
Quick Check
1) Find the area of a regular hexagon given its perimeter is \(60\) meters.
2) Find the area of a regular decagon given its radius is \(3.9\) ft and the length of each side is \(2.4\) ft.
3) Find the area of a regular octagon given its perimeter is \(88\) inches.
Quick Check Solutions
1) Find the area of a regular hexagon given its perimeter is \(60\) meters.
2) Find the area of a regular decagon given its radius is \(3.9\) ft and the length of each side is \(2.4\) ft.
3) Find the area of a regular octagon given its perimeter is \(88\) inches.
Quick Check Solutions