**Regular Polygons and Archimedes' Computation of pi**

Any regular polygon can be inscribed in a circle. This is intuitively obvious, so we will not try to "prove" it. The center of the circle for a polygon with an even number of sides is the intersection of any two diagonals, and the center of the circle for a polygon with an odd number of sides is the intersection of any two angle bisectors:

Archimedes used inscribed polygons to approximate the value of π. He started with a regular hexagon with sides that are 1 unit. Then the radius of the circle containing the hexagon is 1 unit (triangle *AOB* is equilateral since its angles are 60^{o}):

Then he constructed *OQ* through the midpoint *P* of segment *AB*:

Notice that triangles *OAP* and *OBP* are congruent by **SSS**, so angle *APO* is a right angle.

Now look at triangles *OAQ* and *OAP*. Since *P* was the midpoint of *AB*, *AP* is ½ unit in length.

By the Pythagorean Theorem, we can find *OP*, and therefore *CP*:

*OP*^{2} + (½)^{2} = 1^{2}

*OP*^{2} = ¾ = 0.75

*OP* = 0.8660254

Since *OQ* is a radius of the circle, its length is 1, so *QP* = 1 – *OP* = 0.1339746.

Now use the Pythagorean Theorem on triangle *APC* to find *AC*:

*AQ*^{2} = (½)^{2} + (0.1339746)^{2} = 0.26794919

*AQ* = 0.51763809

This is the length of a side of the dodecagon (12-sided polygon) inscribed in the unit circle:

The perimeter of the dodecagon is close to the circumference of the circle. This perimeter is 12 times 0.5176389, or 6.2116571. Since pi is the circumference divided by the diameter, we get an approximate value of 3.10582854.

Now Archimedes did not stop there. He next did the same construction using the dodecagon. Let *OQ*' be a radius that intersects side *AQ* of the dodecagon at its midpoint:

Now we have:

*AP*' is half of 0. 0.51763809, or 0.285819. Using the Pythagorean Theorem on triangle *AP*'*O* we find that *OP*' = 0.9659258, so *Q*'*P*' = 1 – 0.9659258 = 0.0340742, and then using the Pythagorean Theorem on triangle *AP*'*Q*', we find that *AQ*' = 0.2610524.

*AD* is a side of the regular 24-gon inscribed in circle *O*, and the perimeter of that polygon is 24(0.2610524) = 6.2652572. This is closer to the circumference of the circle, so now we estimate pi to be 3.132629.

The following table illustrates this procedure carried out to 8 steps, where *n* is the number of sides of the inscribed polygon: