What is the volume of the largest possible cylinder which will fit inside a sphere of radius $1\text{m}$?

This is the picture to start with:

Call the radius and height of the cylinder $r$ and $h,$ and try to eliminate $r$ or $h$ from

It will help to consider this triangle:

Notice that the hypotenuse is a diameter of the sphere.

The solution can be given in the form

for prime numbers $a,b$. Give the value of $a^b$.