Today, March 14, is “Pi Day.” The date matches the first three digits of pi, 3.14. On the Nerd Calendar, Pi Day is even more important than Festivus or G.H. Hardy‘s birthday.
Pi is the ratio between a circle’s circumference and its diameter. You probably remember it from the formula to calculate the area of a circle:
Area of a circle = Pi times the circle’s radius squared
In honor of Pi Day, let’s look at the Ancient Egyptian method to calculate the area of a circle. It’s from the Rhind Papyrus, which is one of our main sources of information about Egyptian mathematics of that era. Here’s how they did it:
- Measure the diameter of the circle.
- Draw a square with sides whose length is 8/9ths the diameter of the circle.
- Calculate the area of the square.
- Within the limits of the Ancient Egyptians’ ability to measure it, the area of the square is the same as the area of the circle.
For example, suppose that the circle’s diameter is 9 cubits. Then the square’s sides would be 8 cubits and the square’s area would be 8 x 8 = 64 square cubits.
Compare that to the modern formula for the area of a circle, which is pi (3.14159…) times the radius squared.
The radius is half of the diameter, so the radius of the same circle is 4.5 and the square of the radius is 20.25. Plugging it into the modern formula, you get:
3.14159 times 20.25 = 63.618,
very close to the Ancient Egyptian answer of 64. The Ancient Egyptian method gives a value of pi that’s about 3.16, close to the modern value of 3.14159…
Pi is an irrational number, which means you can’t write it as a ratio of whole numbers. That annoyed the Pythagoreans, Ancient Greek mathematicians who based their whole view of reality on ratios of whole numbers. They knew that irrational numbers existed, but they tried not to think about them too much.