Find distance based on angle

Question

I would like to know the formula to find at which distance $x$ I must place a camera at $C$ so $d$ is of certain width:

see here

In other words, I want to find $x$, knowing $\theta$ and $d$. (I know that $d = \tan\theta$ but it doesn't help).

Answer 1

hint

You should have

$$\frac{d}{2}=x\tan(\frac{\theta}{2})$$ or

$$\tan(\frac{\theta}{2})=\frac{d}{2x}$$

thus

$$\tan(\theta)=\frac{\frac dx}{1-\frac{d^2}{4x^2}}=\frac{4xd}{4x^2-d^2}$$ now solve for $x$.

Answer 2

If you drop the perpendicular from $C$ to the side of lenght $d$, you will have :

     /|
    / |  d/2
   /  |
C /_a_|
   x

where the angle $a$ is equal to $\theta/2$, so $\tan(\theta/2)= \frac{(d/2)}{x}$ and then:

$$x =\dfrac{1}{2}\cdot d\cot\left(\frac{\theta}{2}\right) $$