Let A be the line that goes from vertex N or P and touches line R. Line A is perpendicular to R, since R is parallel to edge NP.
At point M, the height is x/2, since the triangle is equilateral. Using Pythagoras' Theorem, where x/2 is opposite, x is hypotenuse, and A is adjacent. We have A = (x√3)/2, for the new triangle formed with A.
Now you calculate the radius, r, of the circle formed by folding line A into a circle, where NP touches R in parallel. We have r = [(x√3)/2]/2pi = (x√3)/4pi. Note: Line A becomes the circumference.
We have four equal triangles here with line A at three points: N -> R, P -> R, and (x/2) -> M. Therefore, the volume formed by the equilateral triangle is exactly 1/2 (of the full cylinder), given by, V(triangle) = { pi [ (x√3)/4 pi ]^2} x = (3x^3)/16pi, where h = x. I've spaced out the LHS a bit to make it more readable.