As we known, the area of triangle ACD is one sixth of area of hexagon.

Let us try 12-sided, 24-sided, and 48-sided polygons.

12-sided:

The triangle ACD in 12-sided is one twelfth of the .

Known:

Segment AC=1

Segment BC=1

Angle ACD=30

Area of triangle ACD:

Area of :

24-sided:

The triangle ACD in 24-sided is one twenty-fourth of the .

Known:

Segment AC=1

Segment BC=1

Angle ACD=15

Area of triangle ACD:

Area of :

48-sided:

The triangle ACD in 48-sided is one forty-eighth of the .

Known:

Segment AC=1

Segment BC=1

Angle ACD=7.5

Area of triangle ACD:

Area of :

6-sided:

12-sided:

24-sided:

48-sided:

Then we could make a formula for calculate the area of an n-sided equilateral polygon, which is inscribed in a circle of a radius r.

Segment AC and Segment BC are radius r of circle.

Angle ACD dependent by the n, n decide how many equilateral triangle in circle. So we use n to divide 360 degree (the degree of central angle degree), then we will got the degree of angle ACD.

So the area of equilateral polygon could be:

Estimate of

6-sided:

12-sided:

24-sided:

48-sided:

In 48-sided the result is most close the .

Use trigonometric ratio(s) to find the area of the .

At first we find the area of triangle ACD.

Known:

Segment BC=1

Angle ABC=90 degree

Angle ACB=30 degree

Angle ACD=60 degree

For finding the triangle area of ACD, we have to find out the base segment AD.

Use the Tan to find the segment AB.

Segment AB = Segment BD

So

Segment BC=the radius of circle

So

Then we use the area formula of triangle:

As we known, the area of triangle ACD is one sixth of area of hexagon.

Let us try 12-sided, 24-sided, and 48-sided polygons.

12-sided:

The triangle ACD in 12-sided is one twelfth of the .

Known:

Segment BC=1

Angle ABC=90 degree

Angle ACB=15 degree

Angle ACD=30 degree

Segment AB = Segment BD

So

Segment BC=the radius of circle

So

Then we use the area formula of triangle:

As we known, the area of triangle ACD is one twelfth of area of equilateral polygon.

24-sided:

The triangle ACD in 24-sided is one twelfth of the .

Known:

Segment BC=1

Angle ABC=90 degree

Angle ACB=7.5 degree

Angle ACD=15 degree

Segment AB = Segment BD

So

Segment BC=the radius of circle

So

Then we use the area formula of triangle:

As we known, the area of triangle ACD is one twenty-fourth of area of equilateral polygon.

48-sided:

The triangle ACD in 24-sided is one twelfth of the .

Known:

Segment BC=1

Angle ABC=90 degree

Angle ACB=3.75 degree

Angle ACD=7.5 degree

Segment AB = Segment BD

So

Segment BC=the radius of circle

So

=0.065

Then we use the area formula of triangle:

As we known, the area of triangle ACD is one forty-eighth of area of equilateral polygon.

6-sided:

12-sided:

24-sided:

48-sided:

Then we could make a formula for calculate the area of an n-sided equilateral polygon, which is circumscribed out of a circle of a radius r.

Segment BC is radius r of circle.

Angle ACB dependent by the n, n is number of the equilateral triangles of ACD in circle. So we use n to divide 360 degree (the degree of central angle degree), then we will get the degree of angle ACD. Because angle ACB is one over two of angle ACD.

Estimate of

6-sided:

12-sided:

24-sided:

48-sided:

In 48-sided the result is most close the .

According this two graph, we can see that circumscribed polygon converges faster, because in first graph, we can easily see that when n is 50, the blue line is more close the green line, and for second graph when the n is 5080, the blue line also more close the green line than red line.

Human will never find the exact value of , because is a nonrepeating decimal. Archimedes’ approach is the idea to find the value of . My work is making accurate value of , I had found out the 9 decimal places of accuracy, this method still useful in today, we even could find 100 decimal places of accuracy. But that is not only way to find the value of , we could use the rope to measure the circumference, than find the value of . But it well not be very accurate.