There's a lot more to geometry than the formulas and proofs that fill up textbooks. Geometry is used each day by the scientists and engineers who build the world around us, but it's also used in simpler, more fun ways. If you've ever made a shot on a pool table, you've used a lot of geometry, maybe without ever realizing it.
A Game of Sciences
There are two sciences that rule pool and other billiards games: physics and geometry. Physics determines how the balls will react. For example, hitting the cue ball off center puts some spin on the ball, which causes it to carom off the side of any other ball that it hits. If you're making one of those tough shots where you need to keep the cue ball from following another ball into a pocket, you've seen this physics lesson in action. Hitting the cue ball straight on would cause it to follow the ball into the pocket, leaving you with a scratch.
You could use geometric formulas to describe the exact path a cue ball and other balls will take, but that's a lot less fun than actually playing pool. Still, knowing a bit about physics, especially rotation, can make you a better pool player.
Understanding Angles
What do you see when you look at a pool table? Angles are everywhere. The table is a rectangle; if you took the corner pockets out, you'd have 90 degree angles at each corner.
Along each side of a pool table there are a series of inlaid markings known as diamonds. There are three diamonds on the two shorter rails and six diamonds, three on either side of the side pocket, on the longer rails. These diamonds are all set an equal distance from each other and from the pockets.
These diamonds form an aiming system that's used for bank shots. In pool, there are really only two types of shots: straight shots, where the cue ball travels straight to another ball that travels straight to the pocket, and bank shots, where either the cue ball or the target ball bounces off a rail on its way to a pocket.
To make bank shots, you need to know a lot about angles. Think of a bank shot as a triangle, with the rail that you shoot from as the triangle's base and the path of the ball as the triangle's sides. If you can create the right triangle, you'll sink the ball every time.
Fortunately, you don't need to whip out a protractor and a calculator to make bank shots, but you might want to try measuring some angles with a protractor until you get a good feel for banking the ball.
To use the diamond system, you'll need to think about angles. In this example, we'll use a right triangle to make a bank shot.
Place a ball on a pool table halfway between a corner pocket and a side pocket, lined up with the middle diamond between the pockets, and about six inches from the rail. Next, place the cue ball straight across the table, again about six inches from the rail. The cue ball and the target ball should be in a straight line.
Your target pocket is the corner pocket closest to the cue ball, on the cue ball's side of the table. The shot forms a right triangle, with the rail behind the cue ball forming one side, the straight line between the two balls forming the second side and the path to the pocket forming the hypotenuse.
From geometry, we know that the sum of all of the angles in a triangle is 180 degrees. That makes the math of this shot very easy, even if the shot itself is hard. Since you've got a right triangle, you want to hit the target ball with the cue ball at a 45degree angle.
You can illustrate this on the table using the diamonds along the rails. First, hold your cue so that it's in a straight line over the cue ball and the target ball. Now tilt it at an angle with the tip heading in the same direction as the corner pocket you want to hit. Mark this spot on the rail behind the target ball with a penny. Next, use your cue stick to show the path between the penny and the corner pocket. These angles should match up, and your penny should be centered between the two diamonds next to corner pocket opposite the one you want to hit.
Geometry and the diamond system have shown you precisely where you need to aim to make your shot. Hitting the target ball at a 45degree angle with the right amount of force will bank it into the corner pocket every time.
NonEuclidean geometry evolved from mathematical breakthroughs of the early 19th century, offering scientists a way to illustrate highly complex concepts of physics, space and time. 
Why is geometry important? Whether you're studying science or just playing a game of baseball, geometry is at work.
