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