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There are 5 subsequares and 4 half subsquares making up the octogon, which nearly covers the circle. Each subsquare has an area of , so the total covered by the octogon is . The octogon does not cover 4 little pieces of the circle, but that is nearly made up for by the eight little triangular pieces of the octogon that slop outside the circle. Possibly the scribe that arrived at the approximate expression for the area thought that quantity 7/9*d^2 was too small, rewrote it as and threw in an extra 1/81*d^2 to get . We can compare these approxmations with the formula we know for the area of a circle of diameter d.

> A1 := d -> 7/9.*d^2;
A2 := d -> (d-d/9.)^2;
A3 := d -> evalf(Pi)*(d/2)^2;

> A := d -> [A1(d),A2(d),A3(d)];

To get an approximation for , compute the area of the circle of diameter 2 by the formulas.

> A(2);

The first approximation is and the second is , which is better but not as good as the approximations obtained later by Archimedes. There is no evidence that the Egyptian scribes had mastered the concept of similar figures and so the notion of did not occur to them.

Problems.

1. Note that the fractional approximations of obtained above bracket it, so their average might be a closer approximation to . Is it?

2. Circumscribe a square around a circle and subdivide the square into a 4 by 4 grid of 16 subsquares. Is it possible to choose a polygon approximating the circle whose area is easy to compute and gives a better approximation to than the octogon above?

3. What group of Egyptians might the scribes have spoken to in order to learn about the idea of similar figures?

4. Archimedes obtained and as bracketing approximations of . Find out how he did this.

> (22/7.+223./71)/2;evalf(Pi);

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