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May 3rd, 2019, 07:23 AM   #1
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Help with Arcs

Hello,

I dont know if this goes in Geometry or Trigonometry, so bear with me.

I would like some help with a math experiment I am doing. You know me, I love to experiment. Here is the problem.

Imagine a perfect circle. The radius is 1. Now, divide the circle into six equal triangles. You now have six triangles with equal sides and angles, with an arc on one side.

Obviously, 2*Pi*r, (r being 1,) is 6.283185307...... Divide that by six, because you divided the circle in six, the arc length would be 1.047197551....

Here is the experiment:

If you had the arc, and the base, which is 1, and the angles the arc hit the base, which are 30 degrees each, How would you find the length of the arc?

Any help with this will be appreciated,

Jared
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May 3rd, 2019, 08:25 AM   #2
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Haven't you just found the arc lengths? They're twice the given angle (in radians).
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May 3rd, 2019, 09:52 AM   #3
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@skipjack, thank you VERY much. You just helped me with my experiment. My experiment was that if I had a line, and an arc on top of it, with both angles given, you could find the arc length.

I did not know about radians, to be honest. Thanks for teaching me about it.

Here's what I did.

I turned the straight line into a triangle with an arc on top. Here is how.

The given angle is 30 for where the arc meets the line on both sides. If there were a radius, it would meet the arc at 90 degrees. That means that if there were a triangle formed from that line, it would be 60 degrees each angle, and the sides would all be 1. Thus the radius is one.

Then, I divided 360 by the angle farthest from the arc. The answer was 6.
I divided 2pi by six, then multiplied it by the radius. It gave me the answer.

According to an online radian calculator, the arc length is 1.0472.... if the line it is on is 1 and the angles are both 30.

The way I did it was this:

(2pi / (360 / (180 - ((90 - angle) * 2 )))) * (side / (2cos (90 - angle.)))

Answer would be, according to my calculator, 1.04719755........

You just helped me with my experiment. Thanks!

Jared

Last edited by skipjack; May 3rd, 2019 at 10:37 AM.
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May 3rd, 2019, 10:54 AM   #4
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If the given angle, in radians, is θ, the arc length is θ/sin(θ).
For the angle you gave, sin(θ) = 1/2, so the arc length is 2θ.
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May 4th, 2019, 09:26 AM   #5
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@skipjack, thank you so much. I had no idea there was an easier way to find it then my complicated equation experiment.

Still, it was quite fun coming up with it and figuring it out on my own and getting the same answer. Thanks so much for teaching me.

Happy calculating,

Jared
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