Chapter 11
Section 2: Areas of Regular Polygons and Circles
Today you are going to find the area of regular polygons. A regular polygon is a shape
where all the sides and angles are the same length.
Remind me again, what is a regular polygon? Write the definition below please. 
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As you can guess regular polygons sometimes have different numbers of sides. So we
need to have a general formula for how to find the area of these different types of shapes.
Area of Regular Polygon = ½*P*a
So lets talk about what those letters stand for.
The “P” stands for perimeter.
What does the P stand for? _____________
And the “a” stands for apothem. Now I know what you’re saying, “what the heck is an
apothem, I’ve never ever heard of that.” Well calm down I’m going to tell you.
Here’s the best definition I can come up with for what an apothem is:
The apothem is the segment that connects the center of a regular
polygon and the middle of one of the sides of the shape.
I am SO FORGETFUL! Can you please remind me what an apothem is again?
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In the homework, it will be easy to find where the apothems are because they will always
be dotted lines. Sometimes however it will just be a regular polygon and YOU will have
to draw the apothem.
So to summarize, to find the area of a regular polygon you need to figure out how long
the perimeter is and you need to figure out how long the apothem is. Then you can just
put them into the formula to find the area. Finding those things can be hard through, so
I’m going to teach you how to do that on the next page. Keep reading! 
apothem
B
Example #1
D
A
14
C
FOLLOW THESE STEPS VERY CAREFULLY! (Be sure to remember how we do
problems like these so that you will be awesome at them in the future! )
1. First notice that using the triangle formula would be difficult in this case because
finding the height would be difficult. But since all the sides and angles are the
same (because it’s a regular polygon), we can use the regular polygon formula
from the first page. Write this formula to the side:
2. So we just need to find the Perimeter and the Apothem. Usually in these
problems one of those things is pretty easy to find. Explain below why the
perimeter in this problem is easy to find:
a. Ok, in the space below do the work to find the perimeter:
b. Hopefully you got that the perimeter is 42!!!! If not, raise your hand.
3. Now on to the apothem. Notice that the apothem (the dotted line) helps form a
little right triangle in picture above. Darken this triangle and label the apothem
with the letter “a”. This is what you will need to solve for.
a. Now, here’s the hard part of the lesson. (So be sure you are paying
attention!) You need to figure out how many degrees the angles in that
right triangle are.
b. The first thing you will do is take 360 and divide by the number of sides in
the figure. (So in this example it would be 3) Write this below:
c. You should have gotten 120! This is the measure of ∠ADC. We need to
cut this in half to get the measure of the angle in our right triangle. In this
case the angle would be 60.
d. Once you have that angle use sine, cosine, or tangent to solve for “a”. Do
this below: (Hint: You will have to split the 14 in half because that is the
length of the side of our right triangle.)
e. Make sure that you get that “a” equals about 4.04. If not please raise your
hand and I will help you.
4. Now that you know the “P” and the “a”, put them into the area for a regular
polygon equation and solve for the area. Do this below:
*Make sure you get that A=84.9. I want to see work on your packet!!
I understand that that was a little bit harder than you’re use to, so it will be extremely
important that as you’re going through these problems that you look back at how we did
this example and repeat these steps one at a time. I know that with practice you will be
doing these like pros in no time! 
Area of Circles
A = πr 2
Just in case you’re not sure, the “A” stands for area and the “r” stands for radius.
Circles are pretty simple shapes so let’s just do a quick example.
Example #2 Find the area of just the circle.
40
Since the side of the square is 40, that means the radius of the circle is 20.
Thus, A = π * 20 2
You should get that A=1256.6
That’s really all there is to circles. You probably have already seen them at some point
during your math career so I’m expecting most of you to be experts.
That’s all the new material. Make sure that as you’re working through the practice
problems on the next page that you’re always referring back to the equations and the
examples we’ve done above. They will help you. I promise. 
Practice Problems: (It might be helpful to get a separate sheet of paper to do your work)
Find the area of each regular polygon. Round to the nearest tenth.
Find the area of the shaded region where necessary.
1.
5.
22
15
2.
6.
8
6
6
3.
7.
4
7
5
4.
12
8.
9