Lesson 8
Relating Area to Circumference
Let’s rearrange circles to calculate their areas.
Problem 1
The picture shows a circle divided into 8 equal wedges which are rearranged.
![Two shapes.](https://cms-im.s3.amazonaws.com/SMov22PKgni2NgGgD5L2Ua6P?response-content-disposition=inline%3B%20filename%3D%227-7.3.B.PP.Image.11.png%22%3B%20filename%2A%3DUTF-8%27%277-7.3.B.PP.Image.11.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233808Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=703d01d87ecbcadaed1e57faaddd61501c8a13e78628b32be71f569e8506a149)
The radius of the circle is \(r\) and its circumference is \(2\pi r\). How does the picture help to explain why the area of the circle is \(\pi r^2\)?
Problem 2
A circle’s circumference is approximately 76 cm. Estimate the radius, diameter, and area of the circle.
Problem 3
Jada paints a circular table that has a diameter of 37 inches. What is the area of the table?
Problem 4
The Carousel on the National Mall has 4 rings of horses. Kiran is riding on the inner ring, which has a radius of 9 feet. Mai is riding on the outer ring, which is 8 feet farther out from the center than the inner ring is.
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In one rotation of the carousel, how much farther does Mai travel than Kiran?
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One rotation of the carousel takes 12 seconds. How much faster does Mai travel than Kiran?
Problem 5
Here are the diameters of four coins:
coin | penny | nickel | dime | quarter |
---|---|---|---|---|
diameter | 1.9 cm | 2.1 cm | 1.8 cm | 2.4 cm |
- A coin rolls a distance of 33 cm in 5 rotations. Which coin is it?
- A quarter makes 8 rotations. How far did it roll?
- A dime rolls 41.8 cm. How many rotations did it make?