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One thing was intentionally not covered in the OP.

Without knowledge of the area formula pi.(r squared) how would the AEs have worked out that a circle with an area of pi square units has a diameter of (approximately) 2 units? My reasoning is as follows:

They would likely square the circle from pi = 19/6 by making the denominator a square (81) thus making the numerator a square (256). The sides of the square would be 16/9.

The diameter of a circle with an area of pi square units would have to be greater than the side of the square (16/9) and less that the diagonal (16/9 x SQR2).

Using 17/12 as SQR2 the diameter would lie between 48 and 68 27ths, that is 58/27 or 19/9.

Using 7/5 as SQR2 the diameter would be between 70 and 102 45ths, that is, 86/45 or 17/9.

The average is 2.

So, the ratio between the diameter of a circle with area of pi square units and the area of the same circle squared is in the region of 2 : (16/9 squared), or 1 : (8/9ths squared). So, subtract a ninth from the diameter and square.

Using this formula and pi as 256/81 it follows that, for every circle, the area is the square of the radius multiplied by pi (as 256/81). I imagine that someone would have spotted this at some time.

Edited 1 time(s). Last edit at 18-Jan-19 23:41 by gjb.

Without knowledge of the area formula pi.(r squared) how would the AEs have worked out that a circle with an area of pi square units has a diameter of (approximately) 2 units? My reasoning is as follows:

They would likely square the circle from pi = 19/6 by making the denominator a square (81) thus making the numerator a square (256). The sides of the square would be 16/9.

The diameter of a circle with an area of pi square units would have to be greater than the side of the square (16/9) and less that the diagonal (16/9 x SQR2).

Using 17/12 as SQR2 the diameter would lie between 48 and 68 27ths, that is 58/27 or 19/9.

Using 7/5 as SQR2 the diameter would be between 70 and 102 45ths, that is, 86/45 or 17/9.

The average is 2.

So, the ratio between the diameter of a circle with area of pi square units and the area of the same circle squared is in the region of 2 : (16/9 squared), or 1 : (8/9ths squared). So, subtract a ninth from the diameter and square.

Using this formula and pi as 256/81 it follows that, for every circle, the area is the square of the radius multiplied by pi (as 256/81). I imagine that someone would have spotted this at some time.

Edited 1 time(s). Last edit at 18-Jan-19 23:41 by gjb.

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