Volume and surface area of similar shapes
By Adam Spencer on June 3, 1998 :

I am studying from a book at home and have become confused and need help. The question is - the ratio of the volumes of two similar cylinders is 8 : 27, what is the ratio of their surface areas?

I know how to calculate the volume of a cylinder and the surface area but need advice on the above.

By Gareth McCaughan on June 3, 1998 :
One preliminary thing. It's possible that you don't know what `similar' means here. It means `the same shape'; so for instance if two cylinders are similar then the ratio between their radii and the ratio between their heights are equal.

Now, as to the question... I suggest you think about the following questions.

  1. The ratio of the heights of two similar cylinders is 1:2. What is the ratio of their volumes? (If you can do that, try it for other ratios until you see what's going on.)
  2. The ratio of the surface areas of two similar cylinders is 1:9. What is the ratio of their heights? (You should be able to do this without much pain once you've done question 1.)
  3. The ratio of the heights of two similar cylinders is 1:2. What is the ratio of their surface areas? (Again, keep doing these until you spot the pattern.)
Once you can answer questions like 2 and 3, you can solve the problem you actually have: you've been told the ratio of volumes, so you can work out the ratio of heights, so you can work out the ratio of surface areas.

Does that help?

By The Editor :

It's worth adding that the same sort of thing happens if you are dealing with two similar cuboids, or cones, or pyramids, or anything else you care to mention.