# Circular Reasoning in Ancient Egypt

In grade school, we all learned how to calculate the area of a circle:

$$A = \pi r^2.$$

This formula dates back to Archimedes and the mathematics of ancient Greece, but the Greeks were not the only ancient civilization to tackle such problems.

Other ancient cultures – including the ancient Babylonians, Egyptians, Indians, Chinese, etc. – also developed various mathematical insights, independently from the ancient Greeks. And they often approached mathematical questions in different ways from their Greek counterparts. For example, the ancient Egyptians seem to have been very practical-minded in their approach to mathematics: they were content with good approximations (rather than exact solutions) and did not feel the need to furnish rigorous “proofs” of the formulas/techniques they used.

A nice example of this “ancient Egyptian” style of mathematics is the method for computing the area of a circle that’s given in the Rhind Papyrus (c. 1550 BC, roughly 1300 years(!) before Archimedes). In essence, the text says that to calculate the area of a circle one should “multiply the diameter by 8/9 and then square the result”. In modern notation this is saying:

$$A = \left(d\cdot\frac{8}{9}\right)^2$$

and, using the fact that $d = 2\cdot r$, this is equivalent to:

$$A = \left(2 \cdot r \cdot \frac{8}{9}\right)^2 = \left( \frac{16}{9} \right)^2 \cdot r^2$$

or, put another way, they (implicitly) assumed that

$$\pi \approx \left( \frac{16}{9} \right)^2 = 3.16049\ldots,$$

which is only about $0.6\%$ larger than the true value of $\pi$. This is not as good as Archimedes’ estimate of:

$$\pi \approx \frac{22}{7} = 3.142857\ldots,$$

which is only off by about $0.04\%$, but it’s still a respectable approximation.

While we know how Archimedes went about approximating $\pi$, it is not entirely clear how the Egyptian author arrived at the value of 8/9 as the “right” number to use in this formula.

But we can speculate . . .

#### A Hypothetical Derivation

Here is one plausible way that an empirically-minded Egyptian engineer might have gotten to 8/9: Imagine drawing a 9×9 square grid and tracing out a circular arc with radius 9 (representing 1/4 of a circle of radius 9 centered at the bottom-left corner of the 9×9 square):

(A similar diagram could easily be drawn in the sand, say, using a piece of rope of length 9.)

Now, if we say a box is “inside the circle” (i.e., colored blue) when its center (represented by a dot) is contained within the circular arc, then we find that 64 of the 81 boxes are “inside the circle” (blue) and the remaining 17 are “outside the circle” (red). [Alternatively, we could declare that a box is blue if most of its area lies within the circular arc and, provided that the diagram is drawn reasonably accurately, it would not be hard to “eyeball” the figure and arrive at the same coloring of the boxes.]

Next, we can suggestively rearrange 17 red boxes along the outer edge of the 9×9 square: $\rightarrow$

This shows that the quarter-circle covers a $64/81 = (8/9)^2$ fraction of the 9×9 square, and therefore that area of the whole circle would also be $(8/9)^2$ of the 18×18 square containing it (which has a width/height of $d$): $\rightarrow$

That is:

$$A = \left(d\cdot\frac{8}{9}\right)^2.$$

This certainly doesn’t prove that this is how the ancient Egyptians arrived at this formula, but it seems like a plausible approach using tools/techniques that would have been available to them.

#### Postscript

After writing this post, I found a presentation by Steven Pigeon where (on slide 18) he describes essentially the same justification for the constant 8/9. He attributes it to the work of Struve & Turaev.