I have posted a similar question here. What I am interested now is how to derive the approximation of $\pi$ from the following text:
Example of calculating a basket. You are given a basket with a mouth of $4 \frac{1}{2}$. What is its surface? Take $\frac{1}{9}$ of $9$ (since) the basket is half an egg-shell. You get $1$. Calculate the remainder which is $8$. Calculate $\frac{1}{9}$ of $8$. You get $\frac{2}{3} + \frac{1}{6} + \frac{1}{18}$. Find the remainder of this $8$ after subtracting $\frac{2}{3} + \frac{1}{6} + \frac{1}{18}$. You get $7 + \frac{1}{9}$. Multiply $7 + \frac{1}{9}$ by $4 + \frac{1}{2}$. You get $32$. Behold this is its area. You have found it correctly.
Wikipedia states that this text corresponds to the following:
$${\displaystyle {\text{Area}}=\left({\frac {2\times 8}{9}}\right)^{2}\times ({\text{diameter}})^{2}={\frac {256}{81}}({\text{diameter}})^{2}}$$
And that this $${\displaystyle {\frac {256}{81}}\approx 3.16049}$$
is an Egyptian approximation of $\pi$.
Can anyone show how to get this approximation of $\pi$ from this text?