Archimedes method to determine the value of the number p (pi)

Let’s consider regular polygons inscribed in and circumscribed around a circle.

A polygon is inscribed in a circle if its vertices are placed on the circle.

A polygon is and circumscribed around a circle if its sides are tangent to the circle.

In the figure below the red equilateral triangle is circumscribed around the circle. All of its sides are tangent to the circle and the red radii are perpendicular to the triangle sides.

The green equilateral triangle is inscribed in the circle. All of its vertices are placed on the circle.

By the same token, in the figure below the red square is circumscribed around the circle. Its four sides are tangent to the circle and the red radii are perpendicular to the sides of the square.

The green square is inscribed in the circle. Its four vertices are on the circle.

In the figure below the red hexagon is circumscribed around the circle. Its six sides are tangent to the circle and the red radii are perpendicular to the sides of the hexagon.

The green hexagon is inscribed in the circle. Its six vertices are on the circle.

In the figure below the red octagon is circumscribed around the circle. Its eight sides are tangent to the circle and the red radii are perpendicular to the sides of the octagon.

The green octagon is inscribed in the circle. Its eight vertices are on the circle.

As we can easily see, P_n the perimeter of the n sides polygon circumscribed around the circle is larger than the circumference length C. On the other hand, the circumference length C is larger than p_n the perimeter of the n sides polygon inscribed in the circle, that is,

which may be read as “p_n is smaller than C which is smaller than P_n”. In other words, the value of the circumference length C is bounded by the values of the perimeters of both n sided polygons (one inscribed in and the other circumscribed around the circle)

Furthermore, is not difficult to realize that both p_n and P_n get closer to the value of C, as the number of sides of the polygons increases.

Archimedes used this very fact to get an approximate value for p (pi) by computing the perimeters of two n sided polygons, for n = 96, one of them circumscribed around and the other one inscribed in the circle, and got for p (pi)

The value of the number p (pi) can currently be written essentially to any desired precision. The first digits are given by

It is worth noting that the values obtained by Archimedes, over two thousand years ago, have a difference of less than 0.1% (one tenth of one percent) with respect to value presently known.