Two hemispheres have volumes in the ratio acubed over bcubed. What is the ratio of their surface areas?

Question

explain

Answer 1

When you have two objects with the same shapes, two dimensional measures, like areas, change with the square of their dimensions, and three dimensional measure, like volume, change with the cube of their dimensions.

Suppose you had two cubes: one where the length of every edge was one, and one where the length of every edge was 2. The total lengths of all of their edges are 12 and 24, respectively, or in a ratio of 1/2, which is the ratio of the lengths of each edge. Their surface areas are 6 and 24, which gives us a ratio of 1/4, which is 1 squared over 2 squared, or 1/2 squared. Their volumes are 1 and 8, which gives us a ratio of 1/8, which is 1 cubed over 2 cubed, or 1/2 cubed.

This is a general principle which applies to all objects with the same shape. So if two hemispheres have volumes in the ratio a cubed over b cubed, then their linear measures, (say, the radii of their circular faces,) have ratio a/b, and so their surface areas have ratio a squared over b squared.

I hope that helps you out. Also, you might get an answer more quickly if you posted questions like this in the Math subcategory, instead of Alternative.

Answer 2

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