Is there a way of determinine the side lengths of a isosceles triangle knowing its angles and area?

Question

I want to be able to determine the side lengths (or at least one side length) of an isosceles triangle knowing only its surface area and angles. Is this possible?

Answer 1

A simple formula for the area $A$ of a triangle given the lengths of two of its sides $a$ and $b$ and the angle 'between' them $C$ is $$ A = \frac{1}{2} ab \sin C$$ In your situation, you'll have $a=b$ and you know $A$ and $C$.

Answer 2

Let the equal sides be of length $a$ and the third side be of length $c$. Let the angles be $A,B,C$, where $C$ is the angle opposite side $c$.

$$\text{Area}=\frac12a^2\sin C$$ Using this, find $a$.

Now, using the cosine rule,

$$\cos C = \frac{a^2+a^2-c^2}{2a^2}$$ $$\cos C = 1 - \frac{c^2}{2a^2}$$

Using this, and the value of $a$, find $c$.