The height of a regular quadrangular pyramid is h and the side of the base is a. Find the total surface area of the pyramid.

In a regular quadrangular pyramid, the base is a square. The side faces are equal triangles, and the height from the vertex falls to the center of the intersection of the square’s diagonals.
The total surface area of ​​a regular pyramid is calculated by the formula:
Stot = Sbok + Sbase = ½ * P * L + a ^ 2, where Sbase is the area of ​​the square, P is the perimeter of the square, L is the apothem.
P = 4 * a.
Let’s calculate the apothem.
To do this, consider one of the side faces of the pyramid. Let’s call it ASB with apothem SD = L, SO = h, point O – the center of intersection of the diagonals of the square.
Then, from the triangle SOD by the Pythagorean theorem, we find:
SD ^ 2 = L ^ 2 = SO ^ 2 + OD ^ 2 = h ^ 2 + (a / 2) ^ 2 = h ^ 2 + a ^ 2/4
L = (h ^ 2 + a ^ 2/4) ^ (1/2)
Means,
S total = ½ * P * L + a ^ 2 = ½ * 4 * a * (h ^ 2 + a ^ 2/4) ^ (1/2) + a ^ 2 = 2 * a * (h ^ 2 + a ^ 2/4) ^ (1/2) + a ^ 2 = a * (2 * (h ^ 2 + a ^ 2/4) ^ (1/2) + a).