Given that a square pyramid has a
volume of 372 cubic centimeters and a height of 31 centimeters, determine the
perimeter of its base.
So, let’s model our square pyramid
with volume of 372 cubic centimeters. The height of 31 centimeters refers
to the perpendicular height of the pyramid. We’re asked to work out the
perimeter of the base of this square pyramid. That’s the distance all the way
around the outside.
Let’s consider what we know about
the volume of a pyramid. We can recall that the volume of a
pyramid is equal to one-third multiplied by the area of the base multiplied by the
height. This won’t directly help us to work
out the perimeter. But if we could work out the area
of the base, we could then go ahead and work out the perimeter. So, let’s start by filling in the
information that we know into this formula.
This will give us 372 — that’s the
volume of the pyramid — equals one-third times the area of the base times 31, which
is the height of the pyramid. We can simplify the right-hand side
by writing a third multiplied by 31 as 31 over three. We then want to isolate the area of
the base, so we perform the inverse operation to multiplying by 31 over three. And that’s to divide by 31 over
three, which is the same as multiplying by three over 31. So, we have 372 times three over 31
is equal to the area of the base.
We can then evaluate this without a
calculator by noticing that 31 goes into 372 12 times. Meaning that the area of the base
is equal to 12 times three, which is 36 square centimeters. So, now that we’ve worked out the
area of the square on the base, we can use this to work out the length of the sides
and, therefore, to calculate the perimeter.
So, if our side lengths are 𝑥 by
𝑥, this means that the area 𝑥 squared would be equal to 36. Therefore, the length 𝑥 is equal
to the square root of 36, which is six centimeters. And so, the perimeter, which is the
distance around the outside of this square, is equal to six plus six plus six plus
six, which is 24 centimeters.