AREA OF A CIRCLE

CIRCLE

A circle is the set of points in a plane that are equidistant from a given point . The distance from the center is called the radius, and the point is called the center. Twice the radius is known as the diameter . The angle a circle subtends from its center is a full angle, equal to or radians.

The perimeter of a circle is called the circumference, and is given by

We have learned that the area of a parallelogram is the product of its base and its height, and the circumference of a circle with radius is .

To find the area of a circle with radius , divide it into congruent sectors (blue and red divisions) then arrange them as shown below.

As the number of sectors increases, the shape of the rearranged sector is becoming more and more parallelogram-like.

Observe that as the number of sectors increases, the shape of the rearranged sectors is becoming more and more like a parallelogram. In fact, if we can divide the circle into an infinite a number of sectors, it seems that the shape of the rearranged sector is a parallelogram.  Assuming that this is true, then the base of a parallelogram is  πr (Explain why.), and its altitude is r.

Since the area of a parallelogram is bh , we just have to multiply the base of the parallelogram which is πr and its height which is r to find its area. Therefore, the area of the parallelogram, which is equal to the area of a circle, is πr2.

http://curvebank.calstatela.edu/circle/circle.htm