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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

image00.jpg

We know two sides and one included angle.

...read more.

Middle

image60.png

image61.png

24-sided:image50.pngimage50.png

image62.png

image63.png

image64.png

48-sided:image51.pngimage51.png

image65.png

image66.png

image67.png

In 48-sided the result is most close the image21.pngimage21.png.

Use trigonometric ratio(s) to find the area of the image02.pngimage02.png.

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

image69.jpg

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

Use the Tan to find the segment AB.

image70.png

Segment AB = Segment BD

So

image71.png

image09.png

Segment BC=the radius of circle

image10.png

So

image72.png

Then we use the area formula of triangle: image73.pngimage73.png

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

image74.png

image75.png

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

12-sided:

The triangle ACD in 12-sided is one twelfth of the image02.pngimage02.png.

Known:

Segment BC=1

Angle ABC=90 degree

Angle ACB=15 degree

Angle ACD=30 degree

image76.jpg

image77.png

Segment AB = Segment BD

So

image78.png

image09.png

Segment BC=the radius of circle

image10.png

So

image79.png

Then we use the area formula of triangle: image80.pngimage80.png

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

image17.png

image82.png

24-sided:

The triangle ACD in 24-sided is one twelfth of the image02.pngimage02.png.

Known:

Segment BC=1

...read more.

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.

image20.png

Estimate of image21.pngimage21.png

image22.png

image23.png

6-sided:image15.pngimage15.png

image25.png

image26.png

image27.png

12-sided:image16.pngimage16.png

image28.png

image29.png

image30.png

24-sided:image18.pngimage18.png

image31.png

image32.png

image33.png

48-sided:image19.pngimage19.png

image34.png

image35.png

image36.png

In 48-sided the result is most close the image21.pngimage21.png.

image37.png

image38.png

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 image21.pngimage21.png, because image21.pngimage21.png is a nonrepeating decimal. Archimedes’ approach is the idea to find the value of image21.pngimage21.png. My work is making accurate value of image21.pngimage21.png, 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 image21.pngimage21.png, we could use the rope to measure the circumference, than find the value of image21.pngimage21.png. But it well not be very accurate.

...read more.

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