Why does stereographic projection preserve angles but not area?

[dori1123]

Why stereographic projection preserves angles between curves but does not preserve area?

 

[mhazelm]

One way to think about it is that you're mapping to a plane from a sphere. If both angles and lengths were preserved, you'd have the same curvature on each. But clearly the curvature isn't the same on each (try adding angles in a "triangle" on the sphere!). Since angles are preserved, you have to lose the lengths to account for the change in geometry.