Aregular rectangular pyramid has a base and lateral faces that are congruent equilateral triangles. it has a lateral surface area of 72 square centimeters. what is the surface area of the regular triangular pyramid?

A pyramid is regular if its base is a regular polygon, that is a polygon with equal sides and angle measures.
(and the lateral edges of the pyramid are also equal to each other)

Thus a regular rectangular pyramid is a regular pyramid with a square base, of side length say x.

The lateral faces are equilateral triangles of side length x.

The lateral surface area is 72 cm^2, thus the area of one face is 72/4=36/2=18  cm^2.

now we need to find x. Consider the picture attached, showing one lateral face of the pyramid.

by the Pythagorean theorem: 



thus, 



thus:

      (cm^2)

but  is exactly the base area, since the base is a square of sidelength = x cm.

So, the total surface area = base area + lateral area =     cm^2

  cm^2