The pyramid is a regular triangular edge 14 one side of the base 15, find the height.

Consider triangle AKC.

AC = 15 (by condition). Angle C = 60 degrees, because in an equilateral triangle, all angles are equal. AK: AC = sin angle C.

AK: 15 = sin 60.

AK: 15 = √3 / 2.

2AK = 15√3.

AK = 15√3 / 2.

AK is the height, bisector and median of triangle ABC. Point O divides the AK in the ratio 2: 1, counting from the vertex A. Hence, if OK = x, then AO = 2x.

We get: AO + OK = AK.

2x + x = 15√3 / 2.

3x = 15√3 / 2.

x = 5√3 / 2.

That is, OK = 5√3 / 2, then AO = 5√3.

Point M is the top of the pyramid.

AM is the edge of the pyramid, which by condition is equal to 14.

AO = 5√3.

MO is the height of the pyramid to be found.

The AMO triangle is rectangular, which means that we use the Pythagorean theorem to search for the MO:

MO ^ 2 + AO ^ 2 = AM ^ 2.

MO ^ 2 + (5√3) ^ 2 = (14) ^ 2.

MO ^ 2 + 25 x 3 = 196.

MO ^ 2 + 75 = 196.

MO ^ 2 = 196 – 75.

MO ^ 2 = 121.

MO = √121.

MO = 11.

Answer: The height of the pyramid is 11.