How do you find the ratio of the circumference to the radius?

Answer 1

There are several possible methods for approximate results.

Method I:

Draw a large circle with a known radius. For example, with a peg and string in a field, using chalk to mark out the circle. Measure its circumference carefully. Divide the circumference by the radius.

Method II:

In a unit circle inscribe a square, a regular pentagon, a regular hexagon and so on. Since the radius is 1, the circumference of the circle is, itself, the value of ratio that you want.

You can use trigonometry to calculate its perimeter. Consider a polygon with n sides. Each of its sides forms an equilateral triangle at the centre, and the angle at the centre is 360/n degrees. Therefore, it can be shown that the length of each side is 2*sin(180/n). And then, since there are n sides, the perimeter is n*2*sin(180/n) = 2n*sin(180/n). The circumference of the circle is always outside these polygons so the required ratio is always bigger than this.

Next, repeat with escribed regular polygons - polygons outside the circle. This time it is the tangent ratio that comes into play, and the required ratio is less than 2n*tan(180/n).

With decagons, you get 6.1803 < X < 6.4984

with 20 sided figures you get 6.2574 < X < 6.3354

with 96-gon you get 6.2821 < X < 6.2854 (apparently this is how far Pythagoras got).

Even with 360-sided polygon, - working with sin(1/2) and tan(1.2) - the estimate is accurate to only 3 decimal places.

Method III:

Get a wheel (or a cylinder). Measure its radius. Mark a point on its circumference. Roll it along a smooth surface several times. Measure the distance between the first and the last time the point ouches the surface and divide by 2*(the number of time the point touched the surface - 1). The -1 is because the first time length = 0 and so, in a sense it is the 0th time.

Answer 2

It's pi.