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.
Experimentation is fun!
Give him a globe, some string, a ruler, a compass, some paper, and a pencil. Then have him try to map 4 different cities from the globe onto the paper, keeping ALL the distances between them the same as on the globe.