This week your student will learn why circles are different from other shapes, such as triangles and squares. Circles are perfectly round because they are made up of all the points that are the same distance away from a center.

- This line segment from the center to a point on the circle is called the
**radius**. For example, the segment from P to F is a radius of circle 1. - The line segment between two points on the circle and through the center is called the
**diameter.**It is twice the length of the radius. For example, the segment from E to F is a diameter of circle 1. Notice how segment EF is twice as long as segment PF. - The distance around a circle is called the
**circumference**. It is a little more than 3 times the length of the diameter. The exact relationship isC=πd , whereπ is a constant with infinitely many digits after the decimal point. One common approximation forπ is 3.14.

We can use the proportional relationships between radius, diameter, and circumference to solve problems.

Here is a task to try with your student:

A cereal bowl has a diameter of 16 centimeters.

- What is the
*radius*of the cereal bowl?- 5 centimeters
- 8 centimeters
- 32 centimeters
- 50 centimeters

- What is the
*circumference*of the cereal bowl?- 5 centimeters
- 8 centimeters
- 32 centimeters
- 50 centimeters

Solution:

- B, 8 centimeters. The diameter of a circle is twice the length of the radius, so the radius is half the length of the diameter. We can divide the diameter by 2 to find the radius.
16÷2=8 . - D, 50 centimeters. The circumference of a circle is
π times the diameter.16⋅3.14≈50 .

This week your student will solve problems about the area inside circles. We can cut a circle into wedges and rearrange the pieces without changing the area of the shape. The smaller we cut the wedges, the more the rearranged shape looks like a parallelogram.

The area of a circle can be found by multiplying half of the circumference times the radius. Using

Or

This means that if we know the radius, we can find the area. For example, if a circle has radius 10 cm, then the area is about 314 cm2, because

Here is a task to try with your student:

A rectangular wooden board, 20 inches wide and 40 inches long, has a circular hole cut out of it.

- The diameter of the circle is 6 inches. What is the area?
- What is the area of the board after the circle is removed?

Solution:

9π or about 28.26 in2. The radius of the hole is half of the diameter, so we can divide6÷2=3 . The area of a circle can be calculatedA=πr2 . For a radius of 3, we get32=9 . We can write9π or use 3.14 as an approximation of pi,3.14⋅9=28.26 .800−9π or about 771.74 in2. Before the hole was cut out, the entire board had an area of20⋅40 or 800 in2. We can subtract the area of the missing part to get the area of the remaining board,800−28.26=771.74 .

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.