How does atan(1) * 4 equal PI? I needed the PI constant in C++, and I was lead to the answer that:
const PI = atan(1) * 4

Note that despite involving code, I'm asking this from a mathematics perspective.
I have 2 questions about this:


*

*Is this an estimation of PI, or should it give me a large degree of accuracy?

*How does this give PI?


On the second point, if I understand atan correctly, it takes a ratio of two sides, and returns the corresponding angle.
That means that atan(1) is referring a situation with 2 equal length sides.
I think I'm misunderstanding something though, as atan(1) gives me 0.7853981633974483, which seems like a very tiny angle.
If anyone can fill in the holes, it would be greatly appreciated.
 A: You certainly know that $\sin{\frac{\pi}{4}}=\cos{\frac{\pi}{4}}=\frac{\sqrt{2}}{2}$ so one has $\tan{\frac{\pi}{4}}=1$ and therefore $\pi=4\tan^{-1}{1}$
A: I'm including this little gif from Wikipedia as a great way to understand radians.

A: The function $\arctan\colon \mathbb{R}\to (-\frac{\pi}{2},\frac{\pi}{2})$ is the inverse of $\tan$. (for the right domain of definition). As $\tan \frac{\pi}{4} = 1$, this means that $\arctan 1=\frac{\pi}{4}$.
Regarding your question about angles: angles are (in mathematics) measured in radians (in $[0,2\pi)$ or $[-\pi,\pi)$), not in degrees: you should expect a value or order $\pi$ or so, not ranging between $0$ and $360$.
A: Math explanation from a non-math person:
In a right angled triangle if the two short sides are equal, the angle is 45 degrees. 
45 degrees in radians is π/4. (The full circumference is 2πr, 180 degrees is π and 45 degrees is π/4)
sin π/4 = cos π/4 because the two sides are equal. 
tan π/4 = tan 45 = 1.
Arctan(1) is the degree (or radian) which returns a value of 1. So arctan of 1 is either 45 degrees or π/4.
π = 4*arctan(1)
A: This shows geometric explanation for relationship between tan, atan and Pi.
Because horizontal segment AB = 1 and vertical segment BD = 1, angle alpha = 45°. From there you can use atan( BD ) to determine 45° in radiant and take that times 4 to get Pi.

A: You are correct, in all numeric programing languages like fortran, c, c++,  and many others, for a program generalized input line that looks something like:
    print, numeric,  %pi , acos(-1), 4*atan(1)

all return the same numeric value. Here the system stored value %pi may be faster than trig evaluation.
    3.141592653589793, 3.141592653589793, 3.141592653589793

