Image Olympic Stadium

Construction of this stadium requires determination of a circular area.

Image

Area of a Circle

To derive the formula for calculating the area of a circle with radius r, we cut a circle into 4 equal wedges as shown in the picture. Arrange the four wedges in a row, alternating the tips up and down to form a shape that resembles a parallelogram. The reason for changing a circle into a "parallelogram" is because we don't know how to calculate the area of a circle yet. We transform a circle into a shape whose area we know how to compute. As shown, the length of the bumped base (top or bottom) is equal to half of the circumference of the original circle and the length of the other side is equal to the radius r. During this process, no area has been lost or gained so that the area of this newly formed "parallelogram" is the same as that of the original circle. However, this "parallelogram" has bumps on both its top and bottom, so we still don't know how to calculate its area.


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[ 4 wedges ]

[ 8 wedges ]

[ 16 wedges ]

[ Infinity ]

To solve this problem, we cut the original circle into a greater number of equal wedges. As we increase the number, the bumps become smoother and the parallelogram looks more and more like a rectangle. As the number approaches infinity, the bumped "parallelogram" becomes a perfect rectangle, with its width equal to Imager and its height equal to r. As illustrated earlier, The width of this newly formed rectangle equals half of the circumference of the original circle and the height is equal to the radius r. As a result,

Area of Circle = Area of Rectangle = (Imager) r = Imager²

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