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How does one find the side lengths of a right triangle in relation to each other using just the angles? I have all three angles. Is this even possible?

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    $\begingroup$ No, this is not possible. If you scale up your triangle by a constant factor in every side, by similarity the angles stay unchanged. However, the sides don't. $\endgroup$ – chubakueno Aug 17 '13 at 2:13
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    $\begingroup$ You will need one more item of information: the side opposite one of the (specified) angles, or area. In spherical geometry, by contrast, if you know the angles you know the sides, and vice-versa. $\endgroup$ – André Nicolas Aug 17 '13 at 2:23
  • $\begingroup$ @AndréNicolas Can you point me to a good source about spherical geometry and your comment? As my intuition tells me, if I scale up the whole sphere, the angles stay unchanged but the sides don't. This means the sphere has a defined size? Wikipedia's page about the topic is practically useless. $\endgroup$ – chubakueno Aug 17 '13 at 2:37
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    $\begingroup$ The point is that in spherical geometry the sphere radius is fixed. You are correct that if you scale up the sphere the sides scale as well, but given the sphere radius the angles give you the sides. The point is that the radius of the sphere sets the scale, while in the plane there is nothing to set the scale. A simple example is a spherical triangle with all angles $\frac \pi 2$. The sides are then a quarter circumference, or the sphere radius times $\frac \pi 2$ $\endgroup$ – Ross Millikan Aug 17 '13 at 3:01
  • $\begingroup$ This is for a fixed radius of sphere. The Wikipedia article on spherical trigonometry is more relevant. Of course, this has nothing to do with your plane geometry question, where similarity definitely does not imply congruence. $\endgroup$ – André Nicolas Aug 17 '13 at 3:20
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you can only get the ratios of the side lengths to one another... Not the actual lengths of the sides..

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  • $\begingroup$ how does one get this ratio? $\endgroup$ – user1814893 Aug 18 '13 at 15:44
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You can use the Law of Sines to get the ratio between the sides. However, you can't get the actual lengths of the sides because similar triangles will have identical angles but different sides.

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  • $\begingroup$ how would one go about this? Is it like the following: sin(a)*sin(c):sin(b)*sin(c)? $\endgroup$ – user1814893 Aug 17 '13 at 3:07
  • $\begingroup$ because when I do anything along those lines it does not work $\endgroup$ – user1814893 Aug 17 '13 at 3:11

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