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 February 14th, 2009, 09:44 AM #1 Newbie   Joined: Feb 2009 Posts: 2 Thanks: 0 How can I calculate the lengths of all 3 sides? If I have a right angled triangle, and I know the other two angles and the area of the triangle, how can I calculate the lengths of all 3 sides? Something tells me the answer is quite simple, but I'm stumped. It's driving me crazy.
 February 14th, 2009, 10:16 AM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: How can I calculate the lengths of all 3 sides? If a triangle has sides a,b and c with the angles opposite each side given by $\alpha,\beta,\gamma$ respectively, then the area A is given by $A=\frac12a^2\frac{\sin\beta\,\sin\gamma}{\sin\alph a}=\frac12b^2\frac{\sin\gamma\,\sin\alpha}{\sin\be ta}=\frac12c^2\frac{\sin\alpha\,\sin\beta}{sin\gam ma}.$ In the case of a right angled triangle, one of the angles - say $\alpha$ - is a right angle, so $\sin\alpha=1,$ and $\gamma= \frac\pi2-\beta.$ This means that $\sin\gamma=\cos\beta,$ so the expression (where a is the hypotenuse) becomes $A=\frac12a^2\,\sin\beta\,\cos\beta=\frac12b^2\,\co t\beta=\frac12c^2\,\tan\beta.$ Rearrange to find a,b and c.
 February 14th, 2009, 10:23 AM #3 Newbie   Joined: Feb 2009 Posts: 2 Thanks: 0 Re: How can I calculate the lengths of all 3 sides? Thanks so much for that! It really was driving my nuts. I was actually half way there and confused myself.
February 14th, 2009, 03:47 PM   #4
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Re: How can I calculate the lengths of all 3 sides?

Quote:
 Originally Posted by SecretSamurai If I have a right angled triangle, and I know the other two angles and the area of the triangle, how can I calculate the lengths of all 3 sides? Something tells me the answer is quite simple, but I'm stumped. It's driving me crazy.
Here is a simpler approach: a,b,c be sides A and B angles opposite a and b (c is hypotenuse). Q=area.

Q=ab/2. sinA/a = sinB/b. Therefore Q=[sinB/(2sinA)]a^2

 February 14th, 2009, 07:07 PM #5 Global Moderator   Joined: Dec 2006 Posts: 19,168 Thanks: 1640 $\text{Area}\,=\,\frac12ab\,=\,\frac12a^2(b/a)\,=\,\frac12a^2\,\tan{B}.$
 February 14th, 2009, 09:16 PM #6 Member   Joined: Feb 2009 Posts: 34 Thanks: 0 Re: How can I calculate the lengths of all 3 sides? actually you don't need but a single angle and you find those sides
 February 16th, 2009, 04:21 AM #7 Global Moderator   Joined: Dec 2006 Posts: 19,168 Thanks: 1640 No, you started with a single angle (the right angle) and the area. You need another angle to allow the sides to be found.

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