by Tim Lehman and Becky Dragan

Assignment #6

IS SPORTS COMMENTATORS' MATH FUZZY?

After the NCAA decreased of the width between the goal posts, some have argued that extra points would be easier from farther back. Such people maintain it would be wise to take a delay of game penalty to get a "better angle" at the extra point. Common sense implies at some point the angle would begin to decrease but the question remains where does it decrease.

For this write-up, we will consider when the ball is on a hash mark. At the hash mark, the angle of a "good" extra point will be at its least. Thus, if it makes no effect with the ball at the hash mark, it will never make an effect.

Click here for a GSP sketch almost to scale (off by half of a yard or less.) In the sketch, angle (ABC) shows the angle a kick can be and still be good with the new goal posts rule. Angle (DEF) demonstrates the angle for the older, wider goal posts.

Let's look at an exact calculation of the best angle to kick a field goal. In the following chart, x is the distance in yards from the back line where the goal post is. So, the goal line on a football field would correspond to 10 yards in the chart. Y is the distance in yards from the right side of the goal post to the right hash mark. W is the distance in yards from the left side of the goal post to the right hash mark. We then calculate angle theta and then angle phi in degrees. Angle theta is the angle from the right goal post to the kicker to the hash mark on the back line. Angle phi is the angle from the left goal post to the kicker to then along the yard line the kicker is on. Finally, we calculate the angle of the field goal.

 x y w arctan(y/x) arctan(x/w) best angle 2 5 9.16666666 68.1985905136482 12.3080158261064 9.49339366024544 4 5 9.16666666 51.3401917459099 23.5747062171872 15.0851020369028 6 5 9.16666666 39.8055710922652 33.2065703341837 16.9878585735511 8 5 9.16666666 32.0053832080835 41.1120904598102 16.8825263321063 10 5 9.16666666 26.565051177078 47.4895529427554 15.9453958801666 12 5 9.16666666 22.6198649480404 52.6241925245532 14.7559425274064 14 5 9.16666666 19.6538240580533 56.7847485253595 13.5614274165872 16 5 9.16666666 17.3540246362613 60.1908858112672 12.4550895524715 18 5 9.16666666 15.5241109967543 63.0121094351021 11.4637795681437 20 5 9.16666666 14.0362434679265 65.3764352296194 10.5873213024541 22 5 9.16666666 12.8042660652868 67.3801350667535 9.81559886795971 24 5 9.16666666 11.7682889320206 69.0959209305567 9.13579013742269 26 5 9.16666666 10.8855270546587 70.5791790606002 8.53529388474111 30 5 9.16666666 9.46232220802562 73.009176719659 7.52850107231539 35 5 9.16666666 8.13010235415598 75.3236068727629 6.54629077308111 40 5 9.16666666 7.1250163489018 77.092591337807 5.78239231329124 45 5 9.16666666 6.34019174590991 78.4861688236631 5.17363943042703 50 5 9.16666666 5.71059313749964 79.6111421919214 4.67826467057895 55 5 9.16666666 5.19442890773481 80.5376777987316 4.26789329353358 60 5 9.16666666 4.76364169072618 81.3136454249843 3.92271288428948

From the chart, one can see that the widest possible angle is at 6 yards from the goal post. However, the extra point (or any field goal for that matter) is not kicked from within the endzone. So the best place to kick the field goal would be at the goal line (or 5-7) yards back because the field goal is usually kicked from 5-7 yards from the line of scrimmage. It would never make sense to take a delay of game penalty to improve the angle.

We conclude by saying that sports commentators would benefit from having a little high school mathematics.

Click here for the NCAA football rules homepage. If rules change, this information will no longer be current.