Math Background

Lesson: Circumference and Area
Developing the Concept

Now that students have discovered pi as the relationship between the circumference and diameter of a circle, it is time for them to use pi to find the area of a circle. Posting these formulas around the room can be helpful to students.

Materials: one copy of a large circle divided into 8 congruent parts; scissors and rulers for each student

Preparation: Draw a large circle on a sheet of paper. Divide it into 8 equal parts as shown below. Make enough copies for the whole class.

  • Say: In the previous lesson, we examined how to find the circumference of a circle, given its diameter. Today we are going to investigate how to find the area of a circle, given its radius.
    Distribute copies of the circle. Distribute scissors, too.
  • Say: Cut out the circle on the sheet of paper I just gave you. Then cut out each of the 8 sections. Put your scissors down when you have finished so I know you are ready to move on.
  • Say: Now rearrange your 8 pieces like this. (Have them rearrange the 8 pieces as shown below.)
  • Ask: What does this figure look like?
    circle cut into eight sections

    Students should say that it looks something like a parallelogram.

  • Ask:How do I find the area of a parallelogram?
    Elicit from students that the area of a parallelogram can be found by using the formula A = b x h.
  • Ask:What would be the base of the parallelogram?
    Students will probably say that it is the length of the bottom of the figure. You may have to point out that the base b would equal half the circumference of the circle, or b =C over two.Since C = pi d, and d = 2r, we can substitute 2pi r for C to show that b = pi r .Write this on the board. Review the substitutions made in the equation so they are clear to students.
  • Ask:What would be the height? Students should say that it is the radius of the circle. Write the following on the board.
    A = b x h, and since b = pi r and h = r,
    A = pi r x r, or
    A = pi r²
  • Say:Therefore, to find the area of a circle, all we need to know is the radius, because we know that pi times 3.14. Find the area of a circle with a radius of 6 inches.
  • Have a student show how he or she solves the problem on the board. Then have students solve the problems below.
    Three circles with radii

Wrap-Up and Assessment Hints
Provide students with plenty of practice working with the formulas for circumference and diameter before moving on to area. Have students explain to you what substitutions they make in equations and why. By verbalizing their reasoning, both you and your students will be able to assess their understanding of the topics and proceed at a reasonable rate.

Houghton Mifflin Math Grade 6