What is the surface area of the square pyramid?

What is the formula for the total surface area of a square pyramid?

If we assume that the pyramid is regular - that is, the tip is centered over the base - and that you are given base side length x and pyramid height h

then the total area is the area of the base ( x2 ) plus four times the area of each triangular face ( 12xs ) where s is the slant height (the distance from the center of a base-side to the tip of the pyramid).

We can find this by Pythagoras: s=(x/2)2+h2−−−−−−−−−√ ; then the full formula becomes A=x2+2xx24+h2−−−−−−√

If the pyramid is NOT regular, the area (but not the volume) will vary with the tip’s location. We can introduce two more variables, a and b , which are the tip’s fractional distance along each base axis - so for a regular pyramid, a=b=12 .

Then the base area is still the same - x2 - but the four triangular face areas vary:

s1=(xa)2+h2−−−−−−−−−√

s2=[x(1−a)]2+h2−−−−−−−−−−−−−√

s3=(xb)2+h2−−−−−−−−√

s4=[x(1−b)]2+h2−−−−−−−−−−−−√

and the total area is the sum

A=x2+x2(s1+s2+s3+s4)

This gives two points of interest:

dAda is independent of dAdb - that is, moving the tip East or West does not change the area of the North or South faces, and vice-versa (for a pyramid aligned to the cardinal directions);

For a pyramid of a given height, the area is minimized when the pyramid is regular - moving the tip away from center in any direction increases the area.

$4034.40

the perimeter of the field is the sum of the lengths of the four sides:

47 yd + 158 yd + 47 yd + 158 yd = 410 yd

the cost of 3 ft (1 yd) of fence is 3×$3.28 = $9.84.

so the cost of jake's wood perimeter fance is

(410 yd) × ($9.84 /yd) = $4034.40