In a regular triangular pyramid, the height is 12, and the height of the base is 15. Find the total surface area.

Using the formula for the height of an equilateral triangle, we define its side.

AH = BC * √3 / 2.

BC = 2 * AH / √3 = 2 * 15 / √3 = 30 / √3 = 10 * √3 cm.

At point O, the heights of an equilateral triangle ABC are divided in the ratio of 2/1, then OH = AH / 3 = 15/3 = 5 cm.

Then in a right-angled triangle DOH, according to the Pythagorean theorem, DH ^ 2 = DO ^ 2 + OH ^ 2 = 144 + 25 = 169.DH = 13 cm.

Determine the area of ​​the base. Sbn = ВС * АН / 2 = 10 * √3 * 15/2 = 75 * √3 cm2.

Sдсв = ВС * DH / 2 = 10 * √3 * 13/2 = 65 * √3 cm.

The total surface area is: Sпов = Sсн + 3 * Sдсв = 75 * √3 + 3 * 65 * √3 = 270 * √3 cm2.

Answer: The surface area of ​​the pyramid is 270 * √3 cm2.