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How to Find the Area of Regular Polygons

Finding the area of regular polygons is the topic of this geometry math video.

First, the video discusses the parts of a regular polygon: apothem and radius. The center of the polygon is the same as the center of the circumscribed circle. The radius of the polygon is the distance from the center of the polygon to its vertex. The number of radii is determined by the number of sides. The apothem it the perpendicular distance from the center of the polygon to a side.

The first example models how to find the different angles of a regular polygon formed by the radius and the apothem. First, you divide the number in interior angles into 360 degrees. Once you have that you can figure out the other two angles quite easily.

Before applying the the area formula of a regular polygon, the video reviews the formula:

$A = \frac{1}{2}ap$

It is good to note that a = length of the apothem and p = perimeter of the polygon. The perimeter may not be calculated. In that case you multiply the number of sides by the length of each side.

This example is similar to the video, but it is different:

What is the area of a polygon with sixteen 36 in. sides and an apothem of $18\sqrt{3} in.$

$A = \frac{1}{2}ap$

$A = \frac{1}{2}(18 \sqrt{3}(16*36))$

$A = 5184 \sqrt{3} in^2 \approx 8979.0 in^2$

Questions for the comment section:

Comparing example 2T on the video and the problem above, how are the problems alike? How are the problems the same?

Example 3T is finding the area of a regular polygon, but you need to find the length of the apothem using the 30-60-90 triangle relationship.

5 Responses

1. Thanks, Mr. Pi.This is a nice post because I can’t remember this formula anymore.

2. It is hard to remember formulas all the time. I am always asking my students to confirm I cited the correct one in my lectures.

3. That formula can be derived if students understand the concept. But honestly, it seems that sometimes my mind is blocked, that i can’t remember how to derive them, even the easy ones. Hmmm… I think I am getting old. hahaha…

4. well how about if your just givin the apothem how do you solve it

5. There is also another way to find the area of a regular polygon. Knowing the sidelengths, of course, divide the polygon into triangles from the center, forming isosceles triangles. Then, draw in the height of the triangle, thus bisecting the base and the central angle (which equals 360 divided by the number of sides). Do the tangent function to find the height of one triangle. Find the area of the triangle by bh/2, then multiply it by the number of sides of the polygon.