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 Post subject: Musings on triangles and circles
PostPosted: Sun Feb 17, 2013 4:32 am 
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Joined: Sun Nov 23, 2008 2:18 am
90 degreesI have been thinking a lot about triangles and circles lately, and wanted to share some observations I thought were interesting and pose some questiongs that I have not been able to solve myself(or find a webpage via google that my screenreader could adequately read to me).

Start with a unit circle center at the origin.
Draw in it an isoceles(sp?) triangle such that its legs are two radii of the circle, one coinciding with the x-axis, and the uneven side a chord of the circle.
label the angle formed by the two radii and the center as theta.
From the definition of the radian, its obvious that the arc length of the arc corresponding to the chord of such a triangle is merely theta in radians, but is there a formula of the length of this chord in terms of theta?
I have found the chord lengths of a few values of theta:
0 degrees gives a chord length of 0(Triangle degenerates to line segment)
36 degrees gives an chord length of 1/phi.(Acute Golden triangle)
60 degrees gives a chord length of 1(isoceles triangle degenerates to equalateral).
90 degrees gives a chord length of sqrt(2)(45-45-90 triangle)
108 degrees gives a chord length of phi(Obtuse Golden Triangle)
120 gives a chord length of sqrt(3)(based on bisecting the triangle into a pair of 30-60-90 triangles)
180 degrees gives a chord length of 2(Triangle degenerates into line segment).
The range 180-360 produces the same values as the range 0-180, but in reverse order.
Negative values of theta produce the same value as their positive counterparts.
In addition to a general formula for chord length from theta, I am also curious about the following:
What values of theta produce chord lengths of 1/sqrt(2) and 1/(sqrt(3).
the chord lengths produced by thetas of 30, 72, 144, and 150.

Oh, and the question that started me down this line of thinking:
Take a pentagram inscribed within a circle and central pentagon with side length 1. What is the arc length between two points of the pentagram(the corresponding chord length is phi^2).

Since simply answering my questions would make for a very short discussion, fill free to post any other interesting tidbits about triangle, circles, or other geometric shapes.

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 Post subject: Re: Musings on triangles and circles
PostPosted: Sun Feb 17, 2013 1:45 pm 
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Joined: Thu Jan 06, 2005 8:53 pm
Location: Los Angeles
Why, I think you're looking for the Law of Cosines

Sorry I don't have any geometry tidbits at the moment :(


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 Post subject: Re: Musings on triangles and circles
PostPosted: Sat Feb 23, 2013 5:07 am 
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Joined: Sat May 21, 2011 12:54 am
Hmm, interesting as I had been noodling around with finding such things myself lately. I was interested when I saw a question somewhere about how to find how far away a chord has to be from the center so that the length from the chord to the far side of the circle matches the length of the chord itself. After I figured that, I kept finding other things that I wanted to get, such as what areas are on each side of the chord, given the chord length k and the radius r. I got that eventually (needs trigonometry using radians, not degrees below).

Area#1 = (r^2)(arcsin(k/(2r)) - (k(4-(k/r)^2)^(1/2))/(4r))
Area#2 = (pi)(r^2) - Area#1


Last edited by nicknicknick on Sun Feb 24, 2013 2:15 pm, edited 1 time in total.

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 Post subject: Re: Musings on triangles and circles
PostPosted: Sat Feb 23, 2013 5:41 am 
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Joined: Mon Feb 28, 2011 4:54 am
Hi Jeffrey

Quote:
From the definition of the radian, its obvious that the arc length of the arc corresponding to the chord of such a triangle is merely theta in radians, but is there a formula of the length of this chord in terms of theta?

Yes. I think it's length=sqrt(2-2cos(theta)).

This formula certainly works for many of the values you mention (it worked for the ones I checked),

Quote:
What values of theta produce chord lengths of 1/sqrt(2) and 1/(sqrt(3).

A chord length of 1/sqrt(2) is produced by theta=arcos(3/4), or roughly 20.7*

A chord length of 1/sqrt(3) is produced by theta=arcos(5/6) or roughly 16.8*

Quote:
the chord lengths produced by thetas of 30, 72, 144, and 150.

If theta=30*, then chord length=sqrt(2-sqrt(3))
If theta=72*, then chord length=sqrt((5-sqrt(5))/2)
If theta=144*, then chord length=sqrt((5+sqrt(5))/2)
If theta=150*, then chord length=sqrt(2+sqrt(3))

I'm reasonably sure all the above is right. Someone else may have other ideas.

Quote:
Since simply answering my questions would make for a very short discussion, fill free to post any other interesting tidbits about triangle, circles, or other geometric shapes.

i really like what happens to mobius strips when you cut them at various widths.

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