In a regular hexagonal pyramid, the apothem is 15 and the height is 12. Calculate the total surface area of the pyramid.

SABCDEF is a regular hexagonal pyramid.
Sп. = So. + Sb.
1. Consider a triangle SOH: the angle SOH = 90 degrees (since SO is the height), SO = 12 is the leg, SH = 15 (by condition) is the hypotenuse (since it lies opposite the angle equal to 90 degrees). Find the leg HO by the Pythagorean theorem:
HO = √ (SH ^ 2 – SO ^ 2) = √ (15 ^ 2 – 12 ^ 2) = √ (225 – 144) = √81 = 9.
Since SO falls into the center of the circle inscribed in the base of the pyramid, HO is the radius of the circle inscribed in the hexagon ABCDEF.
Find the side of the hexagon:
r = √3 / 2 * t,
where a is the side of the hexagon.
√3t / 2 = 9;
√3t = 18;
t = 18 / √3 = 9√3.
Find the area of ​​the base of the pyramid, that is, the area of ​​the hexagon ABCDEF:
So. = 2√3 * r ^ 2;
So. = 2√3 * 81 = 162√3.
2. Sb. = Pa / 2,
where P is the perimeter of the base and a is the apothem.
P = 6 * t;
P = 6 * 9√3 = 54√3.
Sb. = 54√3 * 15/2 = 810√3 / 2 = 405√3.
3. Sp. = So. + Sb.
Sп. = 162√3 + 405√3 = 567√3.
Answer: Sp. = 567√3.