If the two given pyramids are similar and the volume of the larger pyramid is 160 meters cubed, find the volume of the smaller one.
So we’ve been given two pyramids. And the key piece of information in the question is that they’re similar to each other, which means that all of the corresponding lengths between these two pyramids are in the same ratio. We’ve also been given the volume of the larger pyramid and asked to work out the volume of the smaller pyramid.
Let’s think about how to approach this. We’re given a pair of corresponding lengths, the perpendicular heights of each the two pyramids. They’re five meters and 10 meters. We can use this pair of lengths in order to work out the scale factor for the lengths between the two pyramids.
By dividing the larger length by the smaller, we have that the length scale factor or LSF between the two pyramids is 10 divided by five, which is two. This means that all of the lengths in the larger pyramid are twice as long as the corresponding lengths in the smaller pyramid. Does it follow then that the volume of the larger pyramid is twice as big as the volume of the smaller pyramid?
Well, actually no. Length is a one-dimensional measurement, whereas volume is a three- dimensional measurement. And therefore, the relationship between the lengths and the volumes of similar shapes is not exactly the same. However, it is related. If the length scale factor between two similar shapes is 𝐾, then the volume scale factor or VSF is always 𝐾 cubed.
So as we know the length scale factor for these two pyramids, we can work out the scale factor between their volumes. It’s two cubed, which is eight. What this means then is that the volume of the larger pyramid is not twice as big as the volume of the smaller pyramid, but in fact it’s eight times as big.
So if we want to work out the volume of the smaller pyramid, we need to divide the larger volume by eight. So it’s 160 divided by eight, which is 20. And so we have our answer to the problem: the volume of the smaller pyramid is 20 meters cubed.