Math Woods | Free Mathematics Problems and Concepts

Solution

Solution by Ancient Tree · EN

The area of the octagon is made up of one central medium square, 4 medium squares on the sides, plus 4 half medium squares, for a total of 7 medium squares. Since each medium square is made up of 9 unit areas, we deduce that the area of the octagon is $9\times 7=63$ units.
Adding 1, we obtain $63+1=64$ unit areas.
We need to ask what a unit area is in terms of the diameter of the circle: the key relation is
$S_{U}=\left(\frac{D}{9}\right)^{2}.$ From this we obtain, using question 2, that the area of the disc is approximately
$64\times S_{U}=\frac{64D^{2}}{81}=\left(\frac{8}{9}D\right)^{2}=\left(D-\frac{D}{9}\right)^{2}$ which is the stated formula.
The area of a disc is given by $\pi R^{2}$ , where $R$ is the radius of the disc, so $R=\frac{D}{2}$ ; hence, under the Egyptians' approximation:
$\pi\left(\frac{D}{2}\right)^{2}=\frac{\pi D^{2}}{4}=\left(D-\frac{D}{9}\right)^{2}$ Cancelling $D^{2}$ and then isolating $\pi$ , we find $\pi=4\times \left(\frac{8}{9}\right)^{2}=\frac{256}{81}\approx 3.1605$ . So not quite the right value of $\pi$ , but remarkable for a document written more than a millennium before the ancient Greeks!