Question 1089151
A pyramid with a triangle as a base is a triangular pyramid. Since the base is a triangle and a triangle has three sides, a triangular pyramid has three triangular faces.
Thus, the surface area of a triangular pyramid is the area of all of its faces and base combined. When we have a regular triangular pyramid, all of the faces of the pyramid have the same area. Therefore, the surface area of a regular triangular pyramid can be found by adding the area of the base to 3 times the area of one of the faces. That is, 
{{{SA = A + 3a }}}where A is the area of the pyramid's base, and {{{a}}} is the area of one of the pyramid's faces.

Therefore, our surface area formula becomes:
 
{{{SA = A + 3(1/2)bl}}} 

{{{SA = A + (3/2)bl}}},  where{{{b}}} is the base of one of the pyramid's faces, which is also the length of one of the sides of the pyramid's base, and {{{l}}} is the slant height of one of the pyramid's faces 


if given: the surface area of a regular triangular pyramid with a height of {{{18}}} and base length {{{8}}} 


to find the area of a base triangle, we use the formula 
{{{A = (1/2)bh}}}, where {{{b}}} is the base of the triangle, and {{{h }}}is the triangle's height.

find {{{h }}}is the triangle's height:{{{h=sqrt(18^2-(18/2)^2) =15.6}}}

so, the area of a base triangle is:

{{{A = (1/2)18*15.6}}}->{{{A = 140.4}}}

{{{SA = 140.4+ (3/2)18*l}}}

since slant height {{{l=sqrt(h^2+(8/2)^2)=sqrt(15.6^2+4^2)=16.1}}}

{{{SA = 140.4+ 3*9*16.1}}}

{{{SA = 575.1}}}