Suppose the right pyramid is like the one in the following image:

Right pyramid has a square base . This square base has all equal sides where each side has length . We also know that the height of the right pyramid is which is

Now in sqare , is a right triangle with a right angle.

So by Pythogorean theorem:

But

So

We know that

Now is a right triangle with a right angle.

So:

That's the length of the slopping edge which is

So the area of:

So the area of the triangular face is