##SA=lw+lsqrt(h^2+(w/2)^2)+wsqrt(h^2+(l/2)^2)##

The surface area will be the sum of the rectangular base and the ##4## triangles in which there are ##2## pairs of congruent triangles.

Area of the Rectangular Base

The base simply has an area of ##lw## since it’s a rectangle.

##=>lw##

Area of Front and Back Triangles

The area of a triangle is found through the formula ##A=1/2bh##.

Here the base is ##l##. To find the height of the triangle we must find the slant height on that side of the triangle.

The slant height can be found through solving for the hypotenuse of a right triangle on the interior of the pyramid.

The two bases of the triangle will be the height of the pyramid ##h## and one half the width ##w/2##. Through the we can see that the slant height is equal to ##sqrt(h^2+(w/2)^2)##.

This is the height of the triangular face. Thus the area of front triangle is ##1/2lsqrt(h^2+(w/2)^2)##. Since the back triangle is congruent to the front their combined area is twice the previous expression or

##=>lsqrt(h^2+(w/2)^2)##

Area of the Side Triangles

The side triangles’ area can be found in a way very similar to that of the front and back triangles except for that their slant height is ##sqrt(h^2+(l/2)^2)##. Thus the area of one of the triangles is ##1/2wsqrt(h^2+(l/2)^2)## and both the triangles combined is

##=>wsqrt(h^2+(l/2)^2)##

Total Surface Area

Simply add all of the areas of the faces.

##SA=lw+lsqrt(h^2+(w/2)^2)+wsqrt(h^2+(l/2)^2)##

This is not a formula you should ever attempt to memorize. Rather this an exercise of truly understanding the geometry of the triangular prism (as well as a bit of algebra).