The Area of a Circle

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Why is the area of a circle pi times the square of the radius? The usual definition of pi is the ratio of the circumference of a circle to its diameter, so that the circumference of a circle is pi times the diameter, or 2 pi times the radius. The animation above shows that a circle can be cut and rearranged to closely resemble a parallelogram (with height r and base pi times r) of area pi times the square of the radius. By dividing the circle into more than eight slices, the approximation obtained in this manner would be even better. By dividing the circle into more and more slices, the approximating parallelograms approximate the area of the circle arbitrarily close. This give a geometric justification that the area of a circle really is "pi r squared".

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