How are your visual estimation skills?

A typical 3-ball tennis ball canister is shown on the right.  If you happen to have such a canister, pick it up and perform the following investigation:

Without actually measuring anything, make your best guess as to which is larger: the height of the canister or the circumference of its lid.

Do you have an answer?  Using a piece of string or some similarly flexible, but not stretchable object, check your answer by marking the circumference of the lid on the string, then compare it to the height of the canister.

Most people are surprised by the fact that the circumference of the lid is larger than the height of the can, but in fact, the reason is as “easy as π.”  Here’s a way to think about it (think about it before scrolling down and reading the solution):  The height of the canister is equivalent to 3 ball diameters.  How is the lid’s circumference related to the diameter of a ball?

 

 

 

 

 

As mentioned above, the height of the canister is 3 ball diameters, while the diameter of the lid (or canister) is the same as the diameter of a ball.  Therefore, if D is the diameter of a ball, then the height of the canister is 3D, while the circumference of the lid is πD.  Since π>3, the circumference of the canister is greater than its height.

Why do most people guess wrong?  I think the most feasible reason is that it’s really hard to estimate curved lengths, since it requires the ability to “straighten out” the curve to a segment of equal length.  Our brains just don’t work that way!  Of course, many people just have a difficult time estimating even straight line distances, but that’s a topic for another day (or post)!