Finding Approximate Value of π

We can find the first approximate value of π by using some logic & geometry.

Inscribing a circle inside a square

If the radius of the circle is "r" then the sides of the circumscribing square would each be "2r." Hence the perimeter of the square would be 6r. This would be more than the circumference of the inscribed circle which is 2πr.

Hence 2πr <8r. So π has to be less than 4.

Drawing a regular hexagon inside a circle

A regular hexagon with side as "r" can be drawn inside a circle with a radius "r", using just a compass and an unmarked ruler. The perimeter of this hexagon would be 6r. It would be less than the circumference of the circle which is 2πr.

Hence 2πr > 6r. So π has to be more than 3.

Hence π has to have a value between 3 & 4.

Egyptian Approximation

Egyptians discovered that 64 sound seeds which can be thought of as a 8 X 8 square can approximately rearranged in a circle whose diameter was 9. The area of a circle was πr^2.

Hence 8^2 was approximately X (4.5)^2 which gives the value of π as 3.1605!

Mathematician Marcus Du Sautoy even has a story of how they could have discovered it!

Egyptians used to play a game called Mancala which in Tamil Nadu is called Pallanguzhi. The Mancala board has a series of semi-spherical wholes which have to be filled during the game with spherical seeds. He feels that the above idea may have originated after seeing spherical seeds filling a circular cavity!

Archimedes' Approach

We have indicated the principle of inscribing & circumscribing a circle with a regular polygon to find an approximate value of π. Archimedes increased the accuracy of the approximation by increasing the sides of the polygon.

With a 96-sided polygon, he arrived at a range for π between 3.1408 and 3.1428, which is accurate to two places.

Geometrical Approaches

Some other attempts were made with polygons with a larger number of sides. But the efforts came to a stop with Newton used calculus to come out with an infinite series for calculating π.

The infinite series also gave an hint as to the kind of number which π was.