# Area of a Circle or Regular Polygon

There's a nice way to see why the formula for the of radius R is: * R2.
It has an interesting relationship with the formula for the circumference of a circle, which is 2 * Pi * R (and that is a consequence of the definition of Pi, which is defined to be the ratio of the circumference of a circle to its diameter.)

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So consider a regular polygon, which is an N-sided figure with equal side lengths S and equal angles at each corner. There is an inscribed circle to the polygon that has center C and just barely touches the midpoint of every side. A line from C to the midpoint of a side is called the apothem, and suppose this apothem has length R.

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If you cut the polygon along lines from each corner of the polygon to the center C, you will get a bunch of , each with area (1/2)*(base)*(height). Note that each (base) has length S and the (height) is the length R of the apothem, and there are N such . Thus the total area of the polygon is N*(1/2)*S*R, which to say it an way is:
(1/2) (Circumference of the Polygon) * R

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Now notice that if you let N, the number of sides of the polygon, get larger and larger, the polygon's area approaches the area of a circle of radius R. On the other hand, the circumference approaches the circumference of a circle, so that as N goes to infinity, the above formula approaches:
(1/2) (2 * Pi * R) * R

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which is just Pi*R2, the area of a circle!

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Presentation Suggestions:
Draw a circle and cut it into thin pie wedges. Then help students see that each pie wedge is approximated by a very thin triangle, and as we cut the pie into more and more wedges, this approximation gets better and better and in the limit the approximation becomes equality.

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The Math Behind the Fact:
The process of approximating the area of the circle by slicing it into thin wedges is analogous to the process of integration (in ) to find an area. In the limit this approximation gets better and better. To see the relationship between circumference and area in reverse, where derivatives play a role, see Surface Area of a . See Rolling Polygons for more connections between polygons and circles.

Su, Francis E., et al. “Area of a Circle or Regular Polygon.” Math Fun Facts.

Fun Fact suggested by:
Francis Su

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