Football Problem


Chad Crumley



The football rules in college football were changed a few years ago 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. Note: You will need to find out the width of the uprights and the width of the hash marks . . . make a sketchpad model. Is there any merit to some commentatorsŐ argument to take a penalty in order to have a "better angle" on the field goal kick?


Here is a GSP representation of a collegiate football field:


Let us investigate the largest angle by making a point on one of the hash marks (the top one) and connecting segments to the sides of the upright (<DOG).



Let us do some measurements.  Below you can see an angle of 7.41 degrees is formed when placing the ball on the 50-yard line.



 What about the 25-yard line?  An angle of 13.13 degrees is formed.  This angle is larger and increases the probability of kicking the football through the uprights. 


Here is the angle formed when kicking the football at the 10-yard line.



Notice <DOG keeps getting better.  So, to answer the question, the commentators need to do some geometry explorations like above.  The team should not take any penalty to better the angle.  The maximum angle would be reached at the 7-yard line (which is where kicker attempts the extra point).  <DOG measures about 26 degrees.