Finding all sides of a right triangle from area and a angle

Question

The following calculator can figure out the length of all sides of a right angle triangle using only the area and a angle. How is it doing that..?

http://www.triangle-calculator.com/?what=rt&a=T%3D17.5+A%3D35.538&submit=Solve

Answer 1

Lets say you are given one of the angles and lets call this angle $\alpha$.

Lets call the side opposite $\alpha$, $a$ and the side adjacent to $\alpha$, $b$.

Finally lets call the area of this triangle $A$.

We can then say that:$$\tan(\alpha)=\frac{a}{b}\tag{1}$$

and:$$A=\frac{1}{2}ab\tag{2}$$ From (2) we get:$$b=\frac{2A}{a}\tag{3}$$and if we substitute this into (1) we get:$$\tan(\alpha)=\frac{a}{\frac{2A}{b}}=\frac{a^2}{2A}$$$$\therefore a^2=2A\tan(\alpha)\tag{4}$$You can therefore calculate $a$ from (4) and then get $b$ from (3) and use pytharous to get the remaining side.

Answer 2

One way to define Sin, Cos, and Tan is by sides in a right triangle. The area of a triangle is $1/2$ base times height. In your image this means we can define the area to be $T=1/2ba$. We then also have a equation involving these sides and our angle. $Tan(A)=a/b$. Because $Tan(A)$ and $T$ are constants we have two equations and two unknowns so we can solve for the lengths of $a$ and $b$. From there hopefully you can see that we using similar ideas can solve for the third side and use that the sum of the angles in a triangle is 180 to find the third angle.

Answer 3

Say the area is A and one angle you have given is $\theta$ (angle between sides with length $b$ and $c$) other than the right angle.
Say the length of the sides are $a,b,c$ where $c$ is the hypotenuse.

Now from using properties of triangles, we have the formulae:
$$A=\frac{1}{2}ab\sin 90$$ and $$\frac{a}{\sin \theta}=\frac{c}{\sin 90} \Rightarrow a=c\sin \theta$$ and $$a=b \tan \theta$$

This implies $ab=2A$ i.e.
$$a=\sqrt{2A \tan \theta}$$ $$b=\sqrt{\frac{2A}{ \tan \theta}}$$ $$c=2\sqrt{A\csc 2\theta}$$