What`s the surface area formula for a rectangular pyramid?

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=1/2("base")("height")#.

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 Pythagorean theorem, 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).