Lesson 8Relating Area to Circumference

Learning Goal

Let’s rearrange circles to calculate their areas.

Learning Targets

  • I can explain how the area of a circle and its circumference are related to each other.

  • I know the formula for area of a circle.

Lesson Terms

  • area of a circle
  • squared

Warm Up: Irrigating a Field

Problem 1

A circular field is set into a square with an 800 m side length. Estimate the field’s area.

A yellow square with side length of 800m with a green circle inside touching the edges of the square and a radius shown.

  1. About 5,000 m

  2. About 50,000 m

  3. About 500,000 m

  4. About 5,000,000 m

  5. About 50,000,000 m

Activity 1: Making a Polygon out of a Circle

Problem 1

Your teacher will give you a circular object, a marker, and two pieces of paper of different colors.

Follow these instructions to create a visual display:

  1. Using a thick marker, trace your circle in two separate places on the same piece of paper.

  2. Cut out both circles, cutting around the marker line.

  3. Fold and cut one of the circles into fourths.

  4. Arrange the fourths so that straight sides are next to each other, but the curved edges are alternately on top and on bottom. Pause here so your teacher can review your work.

  5. Fold and cut the fourths in half to make eighths. Arrange the eighths next to each other, like you did with the fourths.

  6. If your pieces are still large enough, repeat the previous step to make sixteenths.

  7. Glue the remaining circle and the new shape onto a piece of paper that is a different color.

Problem 2

After you finish gluing your shapes, answer the following questions.

  1. How do the areas of the two shapes compare?

  2. What polygon does the shape made of the circle pieces most resemble?

  3. How could you find the area of this polygon?

Print Version

After you finish gluing your shapes, answer the following questions.

  1. How do the areas of the two shapes compare?

  2. What polygon does the shape made of the circle pieces most resemble?

  3. How could you find the area of this polygon?

Activity 2: Making Another Polygon out of a Circle

Problem 1

Imagine a circle made of rings that can bend, but not stretch.

A ring of circles in three stages. The first image is a circle made of 5 rings. The second circle shows a cut from an outter point on the circle to the center and is labeled, "The rings are unrolled." The third circle shows 6 rows of the unrolled rings stacked one on top of the other. Starting from the bottom row, each row is shorter than the previous row and is labeled, "The circle has been made into a new shape."

Watch the animation.

open applet in presentation mode

Created in GeoGebra by timteachesmath.

  1. What polygon does the new shape resemble?

  2. How does the area of the polygon compare to the area of the circle?

  3. How can you find the area of the polygon?

  4. Show, in detailed steps, how you could find the polygon’s area in terms of the circle’s measurements. Show your thinking. Organize it so it can be followed by others.

  5. After you finish, trade papers with a partner and check each other’s work. If you disagree, work to reach an agreement. Discuss:

    • Do you agree or disagree with each step?

    • Is there a way they can make the explanation clearer?

  6. Return your partner’s work, and revise your explanation based on the feedback you received.

Print Version

Imagine a circle made of rings that can bend, but not stretch.

A ring of circles in three stages. The first image is a circle made of 5 rings. The second circle shows a cut from an outter point on the circle to the center and is labeled, "The rings are unrolled." The third circle shows 6 rows of the unrolled rings stacked one on top of the other. Starting from the bottom row, each row is shorter than the previous row and is labeled, "The circle has been made into a new shape."

  1. What polygon does the new shape resemble?

  2. How does the area of the polygon compare to the area of the circle?

  3. How can you find the area of the polygon?

  4. Show, in detailed steps, how you could find the polygon’s area in terms of the circle’s measurements. Show your thinking. Organize it so it can be followed by others.

  5. After you finish, trade papers with a partner and check each other’s work. If you disagree, work to reach an agreement. Discuss:

    • Do you agree or disagree with each step?

    • Is there a way they can make the explanation clearer?

  6. Return your partner’s work, and revise your explanation based on the feedback you received.

Activity 3: Tiling a Table

Problem 1

Elena wants to tile the top of a circular table. The diameter of the table top is 28 inches. What is its area?

Are you ready for more?

Problem 1

A box contains 20 square tiles that are 2 inches on each side. How many boxes of tiles will Elena need to tile the table?

Lesson Summary

If is a circle’s circumference and is its radius, then . The area of a circle can be found by taking the product of half the circumference and the radius.

If is the area of the circle, this gives the equation:

This equation can be rewritten as:

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 which is 314 cm².

If we know the diameter, we can figure out the radius, and then we can find the area. For example, if a circle has a diameter of 30 ft, then the radius is 15 ft, and the area is about which is approximately 707 ft².