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If we know the angles and area of the triangle, what would be the formula to find the side lengths? For example, we know the area is 400 ft and the angles are 40, 30, and 110 degrees, how could we find the area?

I've been stumped with this question and could really use a suggestion to help me figure out. Thanks

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    $\begingroup$ the area is $400$ square ft $\endgroup$ – Raffaele Sep 17 '17 at 19:48
  • $\begingroup$ "the area is ... how could we find the area?" Did you mean you want to find the perimeter, or the three side lengths? $\endgroup$ – aschepler Sep 18 '17 at 2:52
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First use the law of sines to find the ratio of all the side lengths, in this case sin(40):sin(30):sin(110). then set one side length as 1 and compute the area via herons formula. Finally scale the side length up by the square root of the known area (400) and the are you just computed. This square root will by the correct side length of the side you choose. Compute the rest of the sides with the ratios from the law of sines.

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Hint...use the formula for the area of the triangle in the form $\frac 12ab\sin C$ to get $ab$, $bc$ and $ca$ then you can get each side...

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  • $\begingroup$ wow i completely forgot about that formula! thanks so much $\endgroup$ – BLoby Blob Sep 17 '17 at 17:44
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    $\begingroup$ This approach leads, if we like, to a formula for each side in terms of the known quantities. For example, $a = \sqrt{2M\sin\alpha\csc\beta\csc\gamma}$ where $M$ is the area of the triangle. $\endgroup$ – Jason Zimba Sep 17 '17 at 18:13
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Use this formula for the area to find the circumradius $R$: \begin{align} S_{\triangle ABC}&=2\,R^2\,\sin\alpha\,\sin\beta\,\sin\gamma ,\\ a&=2\,R\,\sin\alpha,\dots \end{align}

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