by Donny Thurston
In this write-up we will be exploring the mathematics behind kicking a field goal in football. In particular, we will be examing the idea of kicking a field goal from the hash marks on a field, largely because being on the hash marks provides a more difficult angle, and so some strategies have been suggested in order to improve that angle. In particular, we've heard from sportscasters how it may be beneficial for a team to actually take a penalty and move backwards, in order to gain a greater angle to the uprights.
In many respects, this would make sense. If we are coming at an angle from the side (as we will see in some graphics in a moment), there is definitely a spot where getting any closer would actually decrease the visible angle between two stationary points (the two ends of the uprights). In fact, once we get really close, the perceived angle would almost dissapear entirely, until it actually does once you are in line with the goal.
So here we have the question: At what point does it become profitable to "back up" to get a better angle on a field goal when on the hash marks?
Let's explore.
Where is that angle?
So, let's first ask ourselves, where would that point at which the angle starts to shrink as we get closer be? Let's make an image on GSP that we can use as a visual.
Here in this image, we have a simulation of a field goal and the hash marks. The uprights are in yellow, and the end of the field is in light blue. The hash marks are the dashed blue line (and, using hidden circles, are equidistant from the closest end to the field goal), and the red point we see along the hash marks represents the point from which we are kicking the football. The dark blue lines show us the angle that the kicker perceives.
Note that this drawing is not to any scale. We will apply measurements later, once we've worked out some of the mathematics behind the problem. This write-up, and the methods that we use to calculate distances, will not require scale, instead relying on what we have learned.
This drawing tells us very little at the moment. So, lets use GSP to measure the angle, and drag the point around.
Here we see an angle when the football is far from the goal.
Here we see the angle increase appreciably when the football is brought closer...
... and here we observe that our suspicions are correct. When we get "too close" we see that the angle drops again.
So, moving it around a little bit, I find that the "sweet spot" appears to be about here, where the angle is largest. So what can we observe about this point? Lets try to connect our three points with a circle, and observe how that behaves.
In this figure, I placed our same approximate points (from the last several figures), but with circles connecting them, and we observe something interesting. The "best" point, appears to occur when the circle is exactly tangent to the hash marks. It is also the smallest circle of them all. In fact, if we follow each circle, we see that as the angles shrink, the circles get bigger. So here we have an idea, perhaps the greatest angle must occur when our circle is tangent to the hashmarks. Upon reflection, we can see that this makes sense mathematically.
Why the smallest circle is best.
When connecting the three points (the kicking point and the two ends of the uprights), we get a circle. In that circle, the goal posts create a chord, AC. The length of this chord is, naturally, constant.
Let us consider, then, a chord of constant length on circles of various sizes. All chords of equal length within a circle denote an arc of equal size as well. However, as a circle grows larger (as can be seen in the previous figure), a chord of constant length intercepts a proportionally smaller arc. Since degrees are a method of measuring an arc in proportion to the circumference of the circle, it could be said that a chord of a circle of smaller circumference intercepts an arc of larger degree measure than a chord of equal length in a circle of larger circumference. For example, if a chord intercepts an arc that has a degree measure of 20, then a chord of the same length must intercept an arc of less than 20 degrees in a circle with a larger circumference. See the following figure for another look at this concept.
Taken to the extreme, this means that when the circle is sufficiently small so that the chord is actually the diameter of the circle, then the arc measure would be 180 degrees. It cannot become any bigger than this when discussing minor arcs, which is the topic at hand. Therefore, we can see that if the chord is any smaller than the diameter, the arc measure it intersects must also be smaller.
So, why does this matter? It matters because the degree measure of angle on the circumference of a circle, an inscribed angle, is dependent on the measure of the arc it intercepts. Specifically, an inscribed angle is exactly half of the measure of the arc it intercepts, in degrees. Therefore, the greatest measure of an inscribed angle is going to occur when the greatest measure of an arc occurs. We can see this in action here:
Therefore, for a given chord length, the smaller the circle with the chord, the larger the intercepted arc measure will be, and the larger the intercepted arc measure is, the larger the inscribed angle that intercepts that arc will be. Therefore, it is in our interest to create the smallest circle possible.
And since the circle must go through all three points in our diagram, and so must touch the hash marks, the smallest circle we can create is tangent to the hash marks!
So where does that leave us?
So, now that we have determined where the best angle must be, we can plug in some real numbers, and determine where that "sweet spot" really is. Lets go back to our figure.
So, here we have our model with our already-assumed "best" angle. Fortunately for us, the fact that our kicking point, "B" is also the point of tangency with the hash marks makes this very easy. Since the point of tangency is right on the circle, the radius must be from the center of the field (determined by the fact that the center of the circle must be on the perpendicular bisector of the chord AC) to B, or half of the distance between the hash marks.
So, our knowns: A college football field has hash marks which are 40 feet (480 in.) apart. This is the diameter of the circle, so the radius must be 240 in. On a college football field, the goal posts are 18 ft and 6 in. (222 in.) apart. Half of this chord is 111 in., and is relevant to our interest in finding the length between the uprights and the point on the field that the football is being kicked from. Remember that we are not finding the actual distance from the goal posts to the football, but, rather, from the goal posts to the line parallel that represent the yard maker.
See below figure:
We can see from here that we simply can use the pythagorean theorem to solve for x, and determine exactly how many yards away from the uprights that our sweet spot is.
Here are the finished calculations, as well as the angle calculated using trigonometry, and the properties of the central and inscribed angles (not GSP). Note that the drawing is not to scale; it is simply demonstrative of our situation. This is useful in that we can substitute different numbers in for different situations, such as high school football (see below).
Conclusion: So, in college football, the maximum angle that a kicker can have on the goal when on the hash mark is exactly 27.55 degrees, and this occurs 17.73 feet (or 5.91 yards) away from the goal posts. So, if they are ever closer to the goal that this, they should definitely take a penalty and move back. Or should they? Let us remember one final fact. The actual end zone is 10 yards long! Since this is a longer distance that our "sweet spot", a kicker would never actually be that close, let alone even closer!
This means that a kicker kicking from the 1 yard line (or about 7 yard line, as the snap would place him) at the hash marks, will always have a better angle than at any point farther back on the hash marks, because we are always moving away from our "sweet spot". Remember this the next time you hear a commentator mention the possibility of taking a penalty to get a "better angle".
Note that we could still get a better angle by using a play to center the ball, which is a sound strategy, so long as you don't mind losing the down.
High School?
So, what about high school football? Does that have a different answer? Let's look.
A high school field typically has hash marks 53 ft 4 in (640 in) apart, and a goal post width of 23 ft 4 in (280 in). So, using the same mathematics, we find:
Assuming that the high school field still has a 10 yard end zone, then it would still never be advantagous to take a penalty to move back and get a better angle on the goal posts.
