The football rules in college football were changed a few years ago have made the uprights 5 feet
narrower than previously. Many game commentators have harped about how much harder it is to kick field goals from the hash marks. Assume the field goal is attempted from the hash marks. At what yard marker does the kicker have maximum angle to the two uprights. Is there any merit to some commentators argument to take a penalty in order to have a "better angle" on the field goal kick?


We will use GSP for this problem. We will construct a "football field" with appropriately scaled sidelines, hashmarks, and goalposts for one half of a field.

Step 1 is to find the exact dimensions of a football field.

length - 50 yds plus a 10 yd endzone (we will construct just one half the field).

width - 53 1/3 yards or 160 feet

sideline to hashmark - 53 1/3 feet

hashmark to hashmark (between hashmarks) - 53 1/3 feet

distance between goalposts - 18 1/2 feet

placement of goalposts - at end of endzone

Next we consruct a model of the field on GSP. The measurements are fairly accurate.

This shows that the angle available from the hashmark at 25 yards is 9.37 degrees. If you like, click here to see how the angle changes as the ball is placed at different yard marks. As you can see, the widest angle available occurs when the ball is nearest the goal line. As to the question of whether it is advantageous to take a penalty to improve chances of a successful field goal; well, that depends on many things. We will assume that the only factor to consider is angle available.

If the penalty simply moves the ball back some number of yards and the ball is placed back onto the hashmarks, then it is obvous that a penalty would decrease chances of a succesful field goal. However, in professional football, and some conferences in NCAA football, ball placement is changed after a penalty. If the ball is placed back into the center of the field, then it is not so clear which would be best. Which do you think has a better (wider) angle?

- a kick from the hashmarks at the 15 yard line or

- a kick from the center of the field from the 20 yard line (after a 5 yard penalty)?

Well, let's see:

The green angle is before the penalty, the red angle is after a penalty. As you can see, it still does no good to take a penalty! Try experimenting with different starting points and penalty distances yourself by clicking here.

Provided hat the minumum penalty is 5 yards, the only situation in which a penalty gives a better angle is when the ball starts at or very near the goal line on the hashmark, then is moved 5 yards back into the center of the field. This situation is illustrated below.

With the ball at the 5 yard line, the penalty does NOT create a better angle. However, when the ball is at the 2 1/2 yard line:

Above you can see that somewhere between the 5 yard line and the 2 1/2 yard line, the angle improves as the penalty is accepted (provided that the ball is moved to center). The closer to the goal line, the greater the angle improvement. Check it out here.

As you can see, it would be a tough job to be a football coach. Not only would you need to know and understand all the rules of the game, you also need to understand how geometry can be used to assist in making decisions! Even jocks use math!

So if you were a coach, would you take a penalty to improve the angle for your kicker? When would you accept a penalty? When would you kick without a penalty? Why?