In the regular triangular pyramid SABC, point M is the middle of the edge BC, S is the vertex. It is known that AB = 6

In the regular triangular pyramid SABC, point M is the middle of the edge BC, S is the vertex. It is known that AB = 6 and the lateral surface area is 45. Find the length of the segment SM

A pyramid is called regular if a regular triangle lies at its base.
Therefore BC = AC = 6
The side faces of a regular pyramid are isosceles triangles and are equal to each other.
Therefore, the area of an isosceles triangle is BCS = 45/3 = 15.
Since M is the midpoint of the edge BC, then SM is the median, and in an isosceles triangle the median and the height dropped from the corner against the base coincide, so SM is the height of the BCS triangle.
BCS area = BC * SM / 2
15 = 6 * SM / 2
SM = 30/6 = 5
Answer: 5.