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by Quirkie on October 29th, 2009
Such a proof depends on your definition of how you measure the length of the circumference.
One way is to find a series of polygons whose vertices lie on the circle, find the perimeter of the polygons and find the limit that the perimeter approaches.
Suppose you have two circles, one diameter 1, one diameter 2.
You can repeat the polygon process with both circles. At each stage, the polygon for circle 2 has twice the perimeter of the polygon for circle 1. In the end, the circumference of circle 2 is twice that of circle 1, so the ratio of diameter to circumference is the same for both circles. This procedure applies to all possible pairs of circles, so the ratio is pi for all circles.
How do we know the polygon perimeters are in the same ratio as their diameters?
Draw the two polygons with the same center and join the center with straight lines to two adjacent vertices of the larger polygon.
These lines enclose two triangles, one small one large. Each triangle has two sides which are both equal to the radius of a circle, so in each case the triangle is isoceles
The two triangles are similar: they have one angle at the center of the circles in common, and as they are both isoceles the other angles are all the same too.
As the triangles are similar, the opposite sides, and therefore the perimeters are also in the same ratio as the diameters.
What is the most weird way to measure the number π?
by Alexander on October 28th, 2009
| 1 person likes this
Is the graph 4X^2 + 4X + 4y^2 + 24y - 107 = 0 a circle? If so how do I find the radius?
by Amber_F7438 on February 9th, 2012
| 2 people like this
You have parametric equations x = t^2-4 and y = t+2
How is y related to x?
by WarHorseLeBron on July 13th, 2010
| 1 person likes this
What theorem would I have to apply to a quadrilateral inside a semicircle?
by Mr_T7336 on August 7th, 2011
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A circle has radius r and circumference c. Solve this equation exactly for r: c/(2r) = 22/7? *** I have to warn you : this question is of my elder brother and he majors in math at a university.
by Alexander on October 28th, 2009
| 1 person likes this
You're reading What does it take to prove that all circles are similar, that is similar in strict geometric sense? 'Cause if not all circles are similar, then how can we use the same constant π=22/7 for a ratio of circumference to diameter of all circles?
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