Consider a pyramid with quadrilateral base so that all 5 vertices of the pyramid are inscribed in the unit sphere. Which such pyramid has the largest surface area?
I would guess the base is square since for all quadrilaterals in a circle, a square maximizes area. I would also think that the apex lies above the centroid of the square base, but I cannot figure out how to prove it.
This figure may help you visualize the geometry:
Everything is determined by a single length: the distance below the equatorial plane on which the four base corners lie.
Call the distance between the plane $z=0$ and the base plane of the pyramid $h$. Your goal is to find $h$. The area depends upon it.
Basic calculus.