$\begingroup$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
$\endgroup$24 Answers
$\begingroup$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...
$\endgroup$2$\begingroup$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.
$\endgroup$$\begingroup$Use this formula for the area to find the circumradius $R$: \begin{align} S_{\triangle ABC}&=2\,R^2\,\sin\alpha\,\sin\beta\,\sin\gamma ,\\ a&=2\,R\,\sin\alpha,\dots \end{align}
$\endgroup$$\begingroup$- $A$ is the area,
- $x, y, z$ are the three sides,
- $X, Y, Z$ are their opposite angles respectively,
$$x = \sqrt{2A\sin X\over\sin Y\sin Z}$$ $$y = \sqrt{2A\sin Y\over\sin X\sin Z}$$ $$z = \sqrt{2A\sin Z\over\sin X\sin Y}$$
$\endgroup$2
