# Video: Finding the Perimeter of the Base of a Pyramid given Its Volume and Its Height

Given that a square pyramid has avolume of 372 cm³ and a height of 31 cm, determine the perimeter of its base.

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### Video Transcript

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.