# The median of the base of a regular triangular pyramid is 6 cm, the height of the pyramid

**The median of the base of a regular triangular pyramid is 6 cm, the height of the pyramid itself is √5 cm. Find the total surface area of the pyramid.**

An equilateral triangle lies at the base of a regular pyramid.

АН – height, bisector and median of triangle ABC. Point O divides AH in a ratio of 2/1, then OA = 4 cm, OH = 2 cm.

The height of an equilateral triangle is: AH = BC * √3 / 2.

BC = 2 * AN / √3 = 2 * 6 / √3 = 4 * √3 cm.

In a right-angled triangle DOH, according to the Pythagorean theorem, DH ^ 2 = OD ^ 2 + OH ^ 2 = 5 + 4 = 9.

DН = 3 cm.

The area of the base of the pyramid is equal to: Sbn = AH * BC / 2 = 6 * 4 * √3 / 2 = 12 * √3 cm2.

Svsd = ВС * DH / 2 = 4 * √3 * 3/2 = 6 * √3 cm2.

Then Sпов = Sсн + 3 * Sвсд = 12 * √3 + 18 * √3 = 30 * √3 cm2.

Answer: The total surface area is 30 * √3 cm2.