Surface Area Of Rectangular Pyramid

The surface area volition exist 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 but 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/two("base of operations")("height")#.

Here, the base is #l#. To detect the tiptop of the triangle, nosotros must find the slant peak on that side of the triangle.

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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/ii#. Through the Pythagorean theorem, we tin see that the slant height is equal to #sqrt(h^2+(due west/2)^2)#.

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

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

Expanse of the Side Triangles

The side triangles' area can exist 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 expanse of one of the triangles is #1/2wsqrt(h^ii+(l/2)^two)# and both the triangles combined is

#=>wsqrt(h^2+(50/two)^2)#

Total Area

Simply add together all of the areas of the faces.

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

This is not a formula you should always attempt to memorize. Rather, this an exercise of truly understanding the geometry of the triangular prism (likewise every bit a bit of algebra).