Back To Article Deriving the Formula for the Area of a Circle

Brief Description: Students will work individually or in groups to derive the formula for the area of a circle.

Objective: Learners will form a rectangle by partitioning a circle and relate A = bh to A = p r2.

Keywords: Area, height, length, perimeter, circumference, diameter, radius, sector.

Materials: One of the following for each individual or group:

• A large, heavy-paper or cardboard circle, about 12" in diameter.
• Scissors.
• Rulers.
• Colored markers or crayons.
1. Discuss with students practical applications for finding the area of a circle. Explain the problems associated with partitioning a circle into unit squares to find its area. Elicit suggestions on how the area might be determined.

2. Pass out the paper circles, scissors, rulers and colored markers or crayons.

3. Have students draw a diameter (it does not need to be exact), and use two different colors to fill in the resulting semicircles.

4. Instruct students to cut the circle in half along the diameter. Then have them cut each of the resulting semicircles into four equal sectors. There are now a total of eight pieces, four of each color.

5. Ask students to assemble the eight pieces so that they form a shape which resembles a rectangle. Provide them with the hint that the same colors should not touch. (The resulting shape consists of sectors "pointing" in opposite directions, side by side).

6. Solicit suggestions as to how to make the shape more like a rectangle. (This can be achieved by cutting each of the sectors in half, again).

7. Have students cut each of the sectors in half, once more, resulting in a total of 16 equal sectors, eight of each color. Solicit suggestions as to how to make the shape even more like a rectangle. (This can be achieved by cutting each of the sectors in half over and over again). Note: Do not allow students to create more than 16 sectors since they can become unmanageable.

8. Ask students to again assemble the sectors "pointing" in opposite directions, side by side. Make sure that none of the same colors are touching.

9. Ask students to equate the parts of the approximated rectangle to the parts of the original circle. The remainder of the lesson involves the mathematical derivation of the formula for the area of a circle.

• The base, b, of the rectangle is equivalent to half of the circumference. The height, h, of the rectangle is equivalent to the radius, r, of the circle. Therefore, using the formula for the area of a rectangle, A = bh, we get b = c/2 and h = r.
• So the formula for the area of the circle is now A = c/2 � r.
• However, we know that the circumference of a circle is equal to the diameter multiplied by pi (d p) so the formula can now be written as A = dp /2 � r.
• Since the diameter is the same thing as twice the radius (2r), the formula can now be written as A = 2rp /2 � r.
• Simplifying this equation, we arrive at A = r p  � r or A = p r2 as the area of a circle.

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