Finding the length of a side of a triangle

Question

I just took the SAT and was wondering if there is any way to find out the length of a side of triangle when you know the three angles and the area of the triangle.

Answer 1

Yes.

Choose an arbitrary value for a side of the triangle. Use the Law of Sines to find the other two sides. Then use Heron's formula

$$A=\sqrt{s(s-a)(s-b)(s-c)}$$

where

$$s=\frac{a+b+c}{2}$$

to find the area of your test triangle. Or you could use

$$A=\frac 12ab\sin\theta$$

Either way, use that area to rescale your test triangle to the proper size.

You can do this all at once with a formula. I see that @Mann has just done so, though I haven't checked it. You are unlikely to remember the formula, however, and the method I just gave is easy to remember. It also can be generalized to other problems, such as finding the sides of a triangle given the altitudes.

Answer 2

Using trigonometry, you know that

$$S=\frac{a^2\sin\gamma\sin\beta}{2\sin(\gamma+\beta)}$$

when $\;\gamma,\beta\;$ are th angles that enclose the side of length $\;a\;$ .

From the above you can calculate $\;a\;$ ...

Answer 3

you can use the two facts:

(a) the sides of the triangles are proportional to the $sin$ of the opposite angle.

(b) the area of the triangle is half the product of the two sides and the $\sin $ of the included angle.

suppose the proportionality constant is $k.$ then the area of th triangle is $$\frac12k^2 \sin A\sin B\sin C = \text{area} \Delta ABC \to k = \sqrt{\frac{2 area \Delta ABC}{\sin A \sin B \sin C}}.$$