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

1 Answer
Jan 2, 2016

Answer:

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

Explanation:

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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.

mathworld.wolfram.com

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).