It is a known fact that there are rectangular prisms with the same volume but different dimensions. Simply take a 2x2x2 cube and break it into two 1x2x2 prisms and rejoin them to make a 1x2x4 prism. Same volume, different dimensions.
But what about their surface areas? Are they the same?
Check before reading on. For a discussion of how to find the surface
area, click here.
A-ha, then that proves that prisms of equal volumes do not
necessarily have the equal surface areas.
How about the other way around. Do objects with the same surface area have the same volume? We'll illustrate this answer with a counter example again.
Consider a rectangular prism with dimensions 1x2x3. Calculate its surface
area.
Now, calculate the surface area of a 1x1x5 rectangular prism.
Check your answer here.
Do they have the same volume?
For a challenge, try to find all the rectangular prism with a surface area of 38 square units (there are two with integer side lengths).
Check your answer here.
And how about a surface area of 54 square units (with integer side lengths).
Check your answer here.
Do any of these prisms with equal surface areas have equal volumes?