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.
