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Similarity and Volume Ratios - Concept
If two solids are similar, then their corresponding sides are all proportional. The ratio of their surface areas is the side ratio squared and note that the ratios of the areas does not give the actual surface areas. The volume ratio for the two solids is the side length ratio raised to the third power. Again, this is not the solids' volume, only the ratio of the volumes.
We can apply similarity to 3 dimensional solids. Here we have 2 different rectangular prisms. And I'm asking the question, are they similar? Well, let's look at corresponding sides of these two solids. 2 corresponds to 6, 4 corresponds to 12 and 3 corresponds to 9. If that's a constant ratio then these two solids are similar. So let's write our, our ratios. 2 corresponds to 6, 4 corresponds to 12 and 3 corresponds to 9. If you look at all of these ratios, I can reduce them and this one's going to be one third, this one's going to be one third and this last one, is one third.
So since they have a constant ratio between corresponding sides, yes these two prisms must be similar. And they are in the ratio of 1:3. So let's see what that would look like if we apply it to the surface areas.
So the ratio of our one dimensional property right now is 1:3. The ratio of their surface areas is going to be a two dimensional property. So we're going to take one third and we're going to square it. So the ratio of their surface areas is going to be 1:9. If we talk about the ratio of the volumes we are going to take that one dimensional property, and since volume is a 3 dimensional property we're going to cube it. So we're going to do one third cubed. So we're going to cube one and we're going to cube 3, and the ratio of their volumes is 1:27.
So to go between these two, you're going to need to go back to one dimension. Put in different way. If I told you the ratio of their surface areas and asked you for the ratio of their volumes, what I would suggest you do is first, take the square root of your two dimensional property to go to one dimension, and then cube your one dimensional property to get to your three dimensional property.
So once you start playing around with the different dimensions it will be real easy to go between 2 dimensions and one dimensions and 3 dimensions.