I heard about the Moscow Papyrus, which has problems from the ancient Egyptians. It is interesting! Problem ten talks about finding the surface area of a basket with an opening of $4\frac{1}{2}$. This question discusses the problem. Quoting the text from there:
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.
I heard that some say that the question is really talking about the surface area of a cylinder. How can this be? It seems the answer given is 32 - how is this?
Is it because the "basket opening" is talking about the diameter of the basket, and we are assuming the height is the same as the diameter? I am assuming that we are discussing lateral surface, and that we don't care if the answer is off by a half or so.