It was known long ago that , the ratio of the circumference to the diameter of a circle, is a constant. Nearly all people of the ancient world used number for . As an approximation obtained though physical measurements with limited accuracy, it is sufficient for everyday needs.

An ancient Chinese text (周髀算经,100 BC) stated that for a circle with unit diameter, the ratio is .

In the Bible, we find the following description of a large vessel in the courtyard of King Solomon’s temple:

*He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high, It took a line of thirty cubits to measure around it.* (1 Kings 7:23, New International Version)

This infers a value of .

It is fairly obvious that a regular polygon with many sides is approximately a circle. Its perimeter is approximately the circumference of the circle. The more sides the polygon has, the more accurate the approximation.

To find an accurate approximation for , we inscribe regular polygons in a circle of diameter . Let denotes the side’s length of regular polygon with and sides respectively,

Fig. 1

From Fig. 1, we have

It follows that

Substituting (4) into (1) yields

That is,

Further simplification gives

Starting with an inscribed square , we compute from (see Fig. 2). The perimeter of the polygon with sides is .

Fig. 2

Clearly,

.

*Exercise-1* Explain, and then make the appropriate changes:

Hint: (5) is equivalent to