Given that the triangular prisms shown are similar and that the volume of the bigger prism is 2112 cubic inches, determine the volume of the smaller prism.
Since we’re told that these 3D shapes are similar, then we know that these will be the same shape but different sizes. This means that every length in the larger prism is in the same proportion to equivalent lengths in the smaller prism. We can see that the height in the smaller prism is 16 inches and the height in the larger prism is 32 inches. Therefore, to go from the smaller prism to the larger prism, we would double the lengths.
We could say that the linear scale factor here would be to multiply by two. To go from the bigger prism to the smaller prism, we would multiply our lengths by a half.
In the question, we’re asked to find the volume of the smaller prism. And it might be tempting to take our given volume for the larger prism and half it. However, the scale factor of a volume is not the same as the linear scale factor. But it will be calculated by the linear scale factor cubed.
As an aside, if we had been looking at the area of two different two-dimensional shapes in a similar proportion, then the area scale factor would be found by squaring the linear scale factor. In this case, we’re looking at the volume. So we don’t need to worry about the area.
So since we want to find the volume of the smaller prism given the larger prism, then we can use the linear scale factor going in that direction. So our volume scale factor will be calculated by one-half to the third power. Since this is equivalent to a half times a half times a half, then our volume scale factor will be one-eighth. So the volume of our smaller prism will be 2112 times one-eighth. And since 2112 divided by eight is 264, then the volume of our smaller prism is 264 cubic inches.