geometry

Since the triangle is an equilateral triangle, the incenter will be the center of the sphere (label this point C).

The incenter is two-thirds of the distance from a vertex to the midpoint of the opposite side; therefore LO  =  (3/2)·LC.

Since LC = 2m; LO  =  3m

Triangle LOK is a 30-60-90 right triangle, making  KO  =  sqrt(3)·m  and  LK  =  2·sqrt(3)·m.

 

Formula for the volume of a cone:  V  =  (1/3)·pi·r²·h   =   (1/3)·pi·[sqrt(3)·m]²​·3m

Formula for the surface area of a cone:  SA  =  pi·r·L + 2·pi·r²  =  pi·sqrt(3)·m·2·sqrt(3)·m + 2·pi·[sqrt(3)·m]²