Rhind Papyrus

The papyrus measure 6 meters long and 1/3 meter wide, and contains 84 problems relating to unit division, geometric series, binary multiplication, and finding the area of a circle. This gives us a look into how the Egyptians determined their value for pi.

Here's how the problem goes. Take a circle, circumscribed inside a square. Divide the square into nine sections, as in a tic-tac-toe board. Obviously, the circle has an area of less than 9 units. However, it also obviously covers more than 5 units. Halving the corner units produces one half unit each, yielding approximately 7 units total. Taking 1.5 units as the radius of the circle, we find a value of pi of 3.1605.

Problem number 50 of the Rhind Papyrus concerns finding the area of a circular field, and thus gives insight to the ancient Egyptians' estimate of Pi (π).

Ahmes suggests "take away 1/9 of the diameter and square the remainder" to find the area of a circle, so:

a=((8/9)·d)²
a=(8·2r/9)²
a=(16/9)²·r²
a=(256/81)·r²

Combining this with our familiar
a=πr², we find

π=256/81=3.16049...
which is within 1% of the correct value of 3.14159...

According to Mario Livio in his book The Golden Ratio, the papyrus was found at Thebes, bought by Scotsman Henry Rhind in 1858, and now resides in the British Museum. The papyrus contains a table of fractions, with simple names for the unit fractions (1/2, 1/3, 1/4, etc.) followed by 87 mathematical problems (not 84 as stated by Martian_Bob above). The scroll as a whole serves as a sort of calculator's handbook, or as Ahmes puts it "the entrance into the knowledge of all existing things and all obscure secrets."