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## Archimedes's Method

### Inscribed and Circumscribed Regular Polygons

Archimedes was the first to give a scientific method for calculating to arbitrary accuracy. He realized that the perimeter of a regular polygon of n sides inscribed in a circle is smaller than the circumference of that circle, and that the perimeter of a regular polygon of n sides circumscribed around a circle is greater than the circumference of that circle. As n approaches infinity, the two perimeters approach the circumference. In fact, one can think of a circle as a regular polygon with infinitely many sides. The images below demonstrate this concept.

 n = 4 n = 5 n = 8

By finding the perimeters of the two n-gons we can, provided that n is sufficiently large, approximate the value of . Let's derive a general inequality that is similar to the one Archimedes used. First we'll derive the equation for the length of the side of an inscribed regular n-gon. is the angle subtended by one side of a regular polygon at the center of the circle.

 Start with the Law of Cosines. For an inscribed regular polygon, a = b = r. Using the sin half-angle formula.

We can similarly derive the equation for the length of the side of a circumscribed regular n-gon, which yields us with:

We know have the equations for each side of both the inscribed and circumscribed polygons. Let's see what we've figured out:

To find the perimeter of each polygon we just need to multiply the length of each side by the number of sides. The circumference of a circle is equal to 2r. The inequality comparing the perimeters to the circumference is, therefore:

We can divide both sides by 2r, which leaves us with the final inequality:

Of course, Archimedes did not have modern trigonometry when he approximated over 2000 years ago. However, by Pythagorus' Theorem, he knew that for a hexagon, sin was equal to 0.5 and tan was equal to 1/. He was able to show that 3 < < 2 using the regular hexagon. Archimedes built from this by successively doubling the number of sides and using the sin half-angle formula. Through this method he was able to make accurate calculations for a 12, 24, 48, and ultimately a 96-sided polygons. With the regular 96-gon, Archimedes was able show that:

3.140845 < < 3.142858