The Slant Height

Inside every right square pyramid (the apex straight above the centre of the base) there is a right triangle. Its upright side is the height, from the apex down to the centre of the base. Its bottom side runs from that centre out to the middle of a base edge, which is half the base edge. Its hypotenuse is the slant height, down the middle of a face.

This pyramid has base edge 6, so the bottom of the triangle is 3. With height 4, Pythagoras gives ${s}^{2} = {4}^{2}+{3}^{2} = 25$, so the slant height is 5. The small square in the base marks the right angle that makes it work.

That is the rule Slant height: ${s}^{2} = {h}^{2}+{\frac{b}{2}}^{2}$. It links three lengths: the slant height, the height and the base edge. Watch the half: the triangle's bottom is half the base, not the whole of it.