Determine the base side length of a right square pyramid whose height is 12 centimeters and volume is 1,296 cubic centimeters.
It can be helpful to draw a sketch of the information. So here we have a pyramid. And because it’s a square pyramid, that means that the base is a square. The fact that this is a right pyramid means that the apex, or the top of the pyramid, lies above the centroid of the base. The only length measurement information we’re given here is that the height is 12 centimeters. And so in order to work out the base side length, we’ll need to use the fact that the volume is 1,296 cubic centimeters.
We can remember that there is a formula regarding the volume of a pyramid. This formula tells us that the volume is equal to one-third times the area of the base multiplied by ℎ, which is the height of the pyramid. In this question, we are given the volume and the height of the pyramid. What this formula would allow us to do then is simply work out the area of the base. However, if we did use this formula to work out the area of the base, that would allow us to work out the side length of the base. That’s because we know that the base is a square.
Now we can apply this formula and fill in the values that we know. The volume is 1,296, and the height is 12, so we have 1,296 is equal to one-third times the area of the base times 12. On the right-hand side, we can simplify one-third times 12 as four. Dividing both sides by four, we have 1,296 over four is equal to the area of the base. Simplifying on the left-hand side, we have now worked out that the area of the base is 324 square centimeters.
At the start of this question, we recognized that finding the area of the base would allow us to find the side length. This is because we know that the base is a square. To find the area of a square, if we say that the side length of the square is 𝑙, then the area is equal to 𝑙 squared. We have already worked out that the area of the base, that’s the area of the square, is 324 square centimeters. And so if we take the side length on this square to be 𝑙, then 𝑙 squared must be equal to 324. To find the value of 𝑙, we would take the square root of both sides. And the square root of 324 is 18. And we know that that’s going to be a positive value because it’s a length.
So we can give the answer that the base side length of this square pyramid is 18 centimeters.