How do you find the surface area of a rectangular pyramid?

Answer 1

For a rectangular pyramid (which is not a square bottom) you can not use the standard formula of Surface Area = B + 1/2 * P * s, because there is more than one slant height.

A rectangular pyramid is made up of 1 rectangular baseand 4 triangles going up from the base to the top of the pyramid. The surface area is the area of all five parts added together

The first bit is a rectangle so you can find the area of it by multiplying its length times its width.

Now we have four triangles, two of them will have a base which is the length of the pyramid and two will have a base which is the width of the pyramid.

The area of a triangle is (1/2*bh), where b is the base (either length of width of the rectangle) and h is the slant height (distance from the base to the top of the pyramid).

The triangles with base = length and the triangles with base = height will have different slant heights. There will be two triangles of each type so the area of all four triangles will be 2(1/2*ls1) + 2(1/2*ws2) = l*s1 + w*s2

If you have been given both slant heights you have enough information to answer the question at this stage,

You will have SA = l*w + l*s1 + w*s2(where l is length, w is width, s1 is the slant length of the triangles with base l, s2 is the slant length of the triangles with base w)

If you do not have the slant lengths you will have to use the Pythagorean Theorem to find them, this will tell you the slant length of the triangle with base l, will be the square root of (w/2)2 + h2 where h is the height of the pyramid (distance from bottom to top through middle of pyramid) similarly the slant length of the triangle with base w will be the square root of (l/2)2 + h2