# Pi(e) approximations in practice

Tonight the God Plays Dice art department made blondies!

These are supposed to be made, according to the recipe, in a pan which is an eight-inch square. But we have no such thing. We do have a nine-inch circular pan, though. Will that do?

Well, what matters is that the two pans have the same area – and therefore that the same volume of batter will have the same thickness and cook roughly the same. (If you thought I was going to solve some PDEs and work out how the heat transfers, you haven’t been paying attention.)

A nine-inch circle has area $\pi (9/2)^2 = 81\pi/4$ square inches, which is about 63.62. An eight-inch square, of course, has area 64 square inches. Not bad!

What would it take for this approximation to be exactly correct? This would require that $81\pi/4 = 64$ exactly; solving for $\pi$ gives \$\pi = 256/81″, which is often credited as an Egyptian approximation to $\pi$ as it implicitly appears in the Rhind papyrus, an ancient Egyptian document of,problems in mathematics. In fact the setting in which this is established there is almost exactly this one – a circle of diameter 9 and a square of side 8 are said to have the same area. See for example these slides for a history of math class by Bill Cherowitzo.

This isn’t the greatest approximation of $\pi$ – in fact $81\pi$ is about 254.46 – but it has the added “virtue” that 256 is a power of two, and 81 is a power of three. We could write $\pi \approx 2^8/3^4$ – it looks nicer that way, I think.

And because Internet law forbids me from mentioning food without posting a picture of it: