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Investigation of circumference ratio - finding the value of pi.

Extracts from this document...

Introduction

Stoney Li 11g HL math

Explain why the marked angle is 30

In this diagram, the C point is the center of the circle and hexagon. The triangle ACD is one sixth of the hexagon. So the angle ACD is one sixth of central angle. The central angle is 360 degree, which mean the angle ACD is 60 degree.  The segment BC connects the midpoint of segment AB to the point C. So segment AB equal segment BD. Because the segment AC and segment CD are radius of circle, the triangle ACD is isosceles triangle. That mean segment BC divides the triangle ACD to two triangles and this two triangles are equal. Which mean angle ACD had divided to angle ACB and angle BCD. So 60-degree divide by 2, the angle ACB equal angle BCD equal 30 degree.

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

At first we find the area of triangle ACD.

Known:

Segment AC=1

Segment BC=1

Angle ACD=60 We know two sides and one included angle.

Middle  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

Conclusion

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.

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