Ancient Egypt

Area of a Circle

The Ancient Egyptians developed a procedure to determine the area of a circle by subtracting one-ninth from the diameter and squaring the result. It is not known how they did this.

Squaring the circle (creating a square with its area equal to a circle) ideally requires a fraction for π with numerator and denominator both squares, as this would then provide the square root of pi, and thus the length of the sides of the square.

By simple inspection, the value of π can be seen to be three-and-a-bit, and the bit can be taken to be an easy single-digit fraction: being one-fifth, one-sixth, one-seventh, or one-eighth. The first and last cases might be removed from consideration as being too large and too small by close inspection. This leaves two very simple approximations to π : 19/6 and 22/7.

The first is the most promising in providing a square for the denominator, because 9, 36 and 81 (all squares) are related to 6 by multiples of 3. It is fortuitous that 81 / 6 x 19 = 256.5, thereby giving 256/81 as an approximation to π. This means that 16/9ths would be the sides of a square with its area approximately π sq. units. In this manner, the circle is squared.
Circle Squared
The Circle Squared.
It can be appreciated that the four small squares have sides of 8/9ths, which is (1 - 1/9th). The diameter of the circle is two units (having a radius of 1), so the ratio of diameter to area is 2 : (2 - 2/9ths) squared, which when simplified is 1 : (1 - 1/9) squared.

So, it follows that the approximate area of a circle can be determined by subtracting one-ninth from the diameter and squaring. QED.

Note: With a diameter of 9 units, the area would be 64 sq. units, but the area is actually 4.52π = 63.62 sq. units. So, the error is 0.6%.

This might be a route to the area formula of π multiplied by the radius squared. Take any diameter, perform the calculation and divide by 256/81 and the result is always the square of the radius.

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