Exploring the Gudermannian Function

Without to much trouble, we can see that the area of our sector is ϕ/2π (πr^2) \\theta

Now for the fun part. The angle of the hyperbolic sector. Note that we are not solving the problem of finding the area of the hyperbolic sector from its own angle, but rather finding its area in terms of a -> the angle of the circular sector. Really what we want to do is relate the hyperbolic sector angle (ψ) to the circular sector angle (ϕ).

ϕ = arctan(sinh(ψ))

We know that the area of a hyperbolic sector is half the area of it's angle. this means its area is 1/2(arcsinh(tan(ϕ)). "And that is a god place to stop." Bonus: Go watch Micheal Penn's wonderful presentation of these ideas: https://www.youtube.com/watch?v=rypoQgdF5cM