Area and Circumference of a Circle

Goal: I will be able to
Tool Bag
understand the area of a circle.
Here’s How…Notes & Examples
Formulas, equations,
Vocabulary, etc.
PENNIES IN A CIRCLE
Watch this video of putting pennies in a circle.
Guess how many pennies will the big circle on the rug
hold?
Low Guess:
Write a guess that you know will be too low.
High Guess:
Write a guess that you know will be too high.
What information would be useful here?
How big is the large circle?
What if I told you it is 22 inches?
Talk to your
partner.
How can we use smaller circles to figure out how many
pennies are in the big circle?
Today, you are going to work with your partner to see if
you can determine how many pennies are in the big
circle using smaller circles.
Using this sheet, you will fill each circle with pennies to see how many
pennies fit in each circle. Write the number in your notes.
Check your answer with the answer sheet. Now measure each circle.
1
21
52
35
5
12
Create a table to see if there is a relationship between
the size of the circle and the number of pennies.
Circle Size
Number of Pennies
Is it a proportional relationship?
Set up a ratio to see if there is a relationship.
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑃𝑒𝑛𝑛𝑖𝑒𝑠
𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟
=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑃𝑒𝑛𝑛𝑖𝑒𝑠
𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟
4 𝑖𝑛𝑐ℎ 𝑐𝑖𝑟𝑐𝑙𝑒
2 𝑖𝑛𝑐ℎ 𝑐𝑖𝑟𝑐𝑙𝑒
6 𝑖𝑛𝑐ℎ 𝑐𝑖𝑟𝑐𝑙𝑒
3 𝑖𝑛𝑐ℎ 𝑐𝑖𝑟𝑐𝑙𝑒
6 𝑖𝑛𝑐ℎ 𝑐𝑖𝑟𝑐𝑙𝑒
2 𝑖𝑛𝑐ℎ 𝑐𝑖𝑟𝑐𝑙𝑒
21
4
≈ 4 =2
=
5
2
52
6
≈ 4 =2
=
12
3
52
6
≈ 10 = 3
=
5
2
Are these equal? No
Is there a way to make them equal or close to equal?
If you square the ratio of the diameters.
𝐴𝑟𝑒𝑎 1
𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 1 2
= (
)
𝐴𝑟𝑒𝑎 2
𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 2
Using this relationship, how many pennies would you
expect in the large 22 inch circle?
Let’s see how good you were…
The area of a circle is related to the square of its
diameter.
1
𝑨𝒓𝒆𝒂 𝒐𝒇 𝒂 𝑪𝒊𝒓𝒄𝒍𝒆 = 𝜋 𝑑 2
4
𝑨𝒓𝒆𝒂 𝒐𝒇 𝒂 𝑪𝒊𝒓𝒄𝒍𝒆 = 𝜋 𝑟 2
How many pennies will fit around the circle?
Do you think it will be a quadratic relationship?
With your partner, place the pennies around the circles
and write down how many fit around each one.
Let’s see how you did.
1
13
5
9
22
18
Create a table to see if there is a relationship.
Circle Size
Number of Pennies
Is it a proportional relationship?
Set up a ratio to see if there is a relationship.
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑃𝑒𝑛𝑛𝑖𝑒𝑠
=
𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟
2 inch circle
3 inch circle
4 inch circle
5 inch circle
5
= 2.5
2
9
=3
3
13
= 3.2
4
18
= 3.6
5
Is there a relationship? Yes.
It is constant, and the constant is about 3.
Using this relationship, how many pennies would you
expect to around the large 22 inch circle?
Let’s see how close you are.
𝑪𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒐𝒇 𝒂 𝑪𝒊𝒓𝒄𝒍𝒆 = 𝝅 𝒅