How to Prove the Area of a Circle Using a Graph Sheet

Determining the area of a circle easily leads to calculating the value of pi.

A simple way to figure the area of a circle is to draw it on graph paper. The area of the circle would be approximately the number of squares inside the circle times the area of each square. This is only an approximation because the circumference of the circle cuts across some squares. You get a closer approximation if you count the number of partial squares as well as the number of complete squares inside the circle. Doing this leads to figuring out the value of pi rather easily.

Things You'll Need

• Graph paper
• Pencil
• Ruler

Instructions

1. Draw a circle with a one inch radius on graph paper. Count the number of whole graph squares inside the circle. Multiply that number by the size of each square. Count the number of partial squares and multiply the number of partial squares times the size of each square and divide that number by 2. Adding the numbers you obtained from both calculations gives you the approximate area of the circle. A circle with a 1-inch radius has an area of approximately 3.14 square inches.

2. Double the radius, this time drawing a circle with a 2-inch radius. This circle has an area of approximately 12.5 square inches. Double the radius again, drawing a circle with a 4-inch radius. The area of this circle is about 50.25 square inches. Doubling the radius quadruples the area of the circle.

3. Divide the area of the largest circle by the area of the smallest circle: 50.25/3.14=16. The radius of that circle was 4, and 16 is the square of 4. Divide the area of the middle circle by the area of the smallest circle: 12.5/3.14=4. The radius of that circle was 2, and 4 is the square of 2.

4. Put that into a formula a different way. The area of a circle with a radius of 1 multiplied against the square of any radius gives the area of a circle with that radius. The area of the circle with the radius of 1 is a constant and it was given the name pi. Thus we have the formula for the area of a circle: pi times the radius squared.

Tips & Warnings

• Use graph paper with smaller squares to give you a more accurate count of the area of a circle with a radius of 1 inch.

References

• Plus Magazine: What is the Area of a Circle
• APEC: Area of the Circle Grade 5 (Japan)

Resources

• "Here's Looking at Euclid: A Surprising Excursion Through the Astonishing World..."; A. Bellos; 2010