TITLE: Why can't you flatten a sphere?
QUESTION [61 upvotes]: It's a well-known fact that you can't flatten a sphere without tearing or deforming it.  How can I explain why this is so to a 10 year old?
As soon as an explanation starts using terms like "Gaussian curvature", it's going too far for the audience in question.
A great explanation would work just as well for a hemisphere as for a sphere, as both have positive curvature.

REPLY [4 votes]: You can discuss curvature with a child. You do it by taking a flat piece of paper, the previously mentioned orange peel and a mustard leaf. From each you cut an annulus and make a transverse slit in it. You show that the flat piece of paper's annulus can be laid flat with no adjustment. You see that the orange peel's annulus opens up when you lay it flat. You explain that that angle is the curvature. The mustard leaf will be more in interesting. When you lay it flat you will find there is excess annulus. You can then explain negative curvature.