Understanding Circles

What is the definition of a circle?

In order to illustrate that the definition of a circle is all points that are the same distance from a point, you can do two activities.

1. Acting out the definition:

Go outside in a large open area.

Have students form a line. Hold a piece of rope that is about 10’ in length.

Have the first person in line hold the rope and start walking forward holding the end of the rope while you stand stationary holding the other end of the rope.

As student continue to follow the first student, they will form a circle around you. They need to write and think about what they did and why that activity formed a circle.

2. Measuring a circle.

Hand out papers with circles on it.

Have students measure the distance from the center to the circle and identify it as the radius.

Ask students to find a place on the circle that is a different measurement than the radius. They will not be able to. What does this mean about a circle?

Finding Pi, using circumference and diameter.

Have students use the recording sheet as they measure 5 items as a group.

Make sure they actually do the measuring and not just guessing or calculating.

Then have them divide the circumference by the diameter.

Have them record their results on a the board. Use the median to see how close they got to the actual value of pi.

Finding the ratio of Circumference to Diameter

The distance around the outside of a circle is called the circumference.

Locate at least 5 round objects and measure the diameter and measure the circumference of each. Record your results in the table below. Be sure to include the units you used in the measuring process.

Object Measured / Circumference / Diameter

Find the area of a circle.

This activity can be done in several ways.

1. The way we did it together was to find the size of r2 and then make a rectangle of sorts out of the circle. Then we placed the squares on the rectangle to see how many fit.

2. You can also do the circle on grid paper and find the area of r2. For instance, with a radius of 5, r2 is the size of 25 units2. Students can color in 25 square units red, then another 25 in blue, and a third in green. This way they can also see that there are 3 + r2 in a circle.

Following are some worksheets for accomplishing these discovery activities.

Finding the Area of a Circle

Draw a radius on this circle.

Make a square that has the radius as a measurement for both the length and the width.

This is called r2, which means the radius squared (or making a square with length of radius).

If I were to fill the area of the circle with radius squares, how many do you think I would need?

What is the area of r2? (How many small squares are in r2?

Using different colors, color in the circle 25 squares in each color and see how many it takes to fill the circle.

What have you just done?

What did you discover about the number of r2s that fit into a circle?

Finding the area of a circle

Area is found using the idea of how many squares can fill a space.

Area of a circle is defined the same way, but there is a relationship between the area of r2 and the area of a circle.

Instructions: Notice the radius square in the circle below. How do you know that it is a square with the dimensions of radius by radius?

Cut out the four radius squares that are to the right of the circle.



Practice Area problems (Leave answers in terms of π for questions 1, 2, and 4)

1.If a side of the square has length 20, what is the area of the circle?

2.What is the combined area of the two circles?

3.If the length of the radius of the circle is 6, what is the area of the square?

4. What is the area of the shaded region?

5. If I double the radius of a circle what happens to the area? Convince me.