Cylinders are flat?!

Questioning whether cylinders are flat seems... bonkers. 
     Flat things aren't round and cylinders are round. These are defining features of what it means to be flat or cylindrical. 
     Hence conflating them when they're essentially opposites seems beyond reasonable. But there are reasons that explain this apparent discrepancy.
     The answer lies in geometry, the study of shapes. (Even though shape is what causes this confusion in the first place!)

Types of Geometry

There are three types of geometry: Euclidean, spherical, and hyperbolic. They can be distinguished by different curvature, parallel lines and triangle degree totals.

Euclidean geometry applies to flat surfaces. They have zero curvature. 
     Parellel lines on Euclidean geometry neither converge nor divurge (that is, they stay parellel). In other words, the distance between parallel lines neither increases nor decreases but stays the same.
     Angles in a triangle on flat geometry adds up to 180 degrees.

Spherical geometry applies to spheres (obviously). 
     They have positive curvature. As such, the surface curls inwards in all directions (stand on any point on the surface and it bends down). 
     Parallel lines converge. This means they will eventually cross paths. The distance between them decreases. Think of longitudinal lines of globes: these are parallel lines that converge at both poles.
     The angles in a triangle on spherical geometry add up to more than 180 degrees.

Hyperbolic geometry applies to saddles. 
     They have a negative curvature. That is, the surface curves in opposite directions (north and south curve up; east and west curve down). 
     Parallel lines divurge. This means they get further and further from each other. The distance between them increases.
     Angles in triangles on hyperbolic geometry add up to less than 180 degrees.

Flat Cylinder?

Cylinders are three-dimensional Euclidean surfaces. As such, all the Euclidean qualifications apply. 
     Parallel lines on cylinders neither converge or divurge. Triangle angles on cylinders add up to 180 degrees. 
     Plus the curvature is zero on a cylinder. So the equations that show curvature equals zero for flat surfaces, whether two-dimensional surfaces like paper or three-dimensional surfaces like cylinders.
     
Zero curvature is obviously the sticking point of the statement 'cylinders are flat'. Definitions of words conflict with definitions of maths.
     Cylinders are intrinsically round like a circle whereas flat things are intrinsically not round. So how can a cylinder be flat? It seems like an obvious contradiction.
     Roll up a flat piece of paper and it becomes a cylinder; unroll a cylinder and it becomes a piece of paper. In both cases, the shape isn't distorted no matter which dimension it's in. Therefore they are the same in geometry.
     Hence the zero curvature of the two-dimensional Euclidean surface applies fully to the three-dimensional Euclidean surface. That is, the flatness of the paper equals the cylinder being flat, too.
     
This is why working with definitions is intresting: sometimes it makes the impossible possible.