This post illustrates an alternative of compute the approximate value of .
We begin with a circle whose radius is , and let represent the side’s length of regular polygon inscribed in the circle with and sides respectively, where
Fig. 1
On one hand, we see the area of as
.
On the other hand, it is also
Therefore,
Or,
where by Pythagorean theorem,
Substituting (2) into (1) gives
That is,
Let , we have
Solving (3) for yields
Since must be greater than (see Exercise 1), it must be true (see Exercise 2) that
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 through 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 represent, the side’s length of regular polygon with and sides respectively, where
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: