The Pythagorean Theorem
Or: DID YOU SEE THAT THROW, IT MADE ME DO MATH
A football field is 53 and 1/3 yards wide.
That number isn’t as well-known as the 120-yard length of the field for several reasons, chief among them that football doesn’t give credit for horizontal yardage gained. A 20-yard gain is a 20-yard gain whether the ballcarrier slices straight through the heart of the defense or reverses field to circumvent it.
The game encourages attacking in vertical lines. Traditional passing schemes are built on reading the field from within the pocket and bailing to the sideline when forced outside a play’s structure.
Designed routes that test the width of the field tend to be short and quick: screens, hitches, flats, fades, maybe some outs if the quarterback has a strong arm and good timing. Throws that take what the defense is willing to allow.
This structure, along with the nature of attempting to hurl an oblong object a long distance with accuracy, created a cardinal rule of quarterbacking: do not, under any circumstances, throw the ball across your body over the middle of the field.
Hi, I’m JJ McCarthy. Welcome to Jackass.
Rules are meant to be broken.
Around the same time I internalized the dimensions of a football field, I learned the Pythagorean Theorem, which tells us how to measure the hypotenuse of a right triangle. Until this weekend, I’d never applied my knowledge of the two to the same subject. I’m not inclined to do math unless I must or I’m truly inspired.
The hash marks in college football are 20 yards from the sideline. The top of the numbers are usually halfway between the hash marks and the sideline. When McCarthy pulled up to throw, he was about halfway between the numbers and the sideline at the Michigan 27-yard line. We’re calling that five yards for the sake of this exercise.
McCarthy’s throw didn’t just reach Daylen Baldwin at the Western Michigan 37, a mere 36 yards downfield using the standard football unit of distance. Baldwin actually zigged to the middle of the field when McCarthy let it fly, then zagged back outside as the ball took him away from his defender by shooting over everyone’s heads at alarming speed. He caught it about five yards inside the hash mark, or 15 yards from the opposite sideline.
Pythagoras tells us that A² + B² = C². We only need to subtract 20 yards (5 + 15) from the width of the field for one side of this right triangle. The second is the vertical distance of the throw: 36 yards. The true distance of the throw is the hypotenuse.
Do the math1 and you get a hair over 49 yards.
Given I’m eyeballing this, ESPN commentator Dan Orlovsky was dead on when saying McCarthy threw it “50 yards on an absolute rope.” Good eye, Orlovsky. Also: hot damn.
McCarthy didn’t break Football Math just to show off. There’s reason to go against the old ways, when coaches schemed to minimize the damage done by their quarterbacks to their own offense, not maximize what they could inflict on the defense.
McCarthy threw Baldwin open. When he set his feet to throw, Baldwin’s defender was right with him. For most quarterbacks, this is not an option.
Baldwin had a half-step over the top, however, and when he took his man to the middle of the field, that left even more room to put the ball over the top and away of a defender who’s playing by the old rules, the right way to play against 99% of college quarterbacks.
By firing the ball 49 yards at an angle, McCarthy gave Baldwin—who, to his credit, made a great adjustment—nearly five yards of separation at the point of the catch.
It turns out throwing across the field is fine, even good—if you can throw it all the way across the field. There’s no way to defend this except hope the opponent doesn’t flex and bend the matrix.
Modern offenses, however, are increasingly reliant on quarterbacks flaunting the outdated rules. In the NFL, it’s been Aaron Rodgers, Patrick Mahomes, Josh Allen, Lamar Jackson, Kyler Murray, and Justin Herbert using their legs to extend plays before their arms make spectacular ones. In college, we’ve seen—well, all of those guys, plus Trevor Lawrence and (sigh) Justin Fields and now a whole incoming generation of quarterbacks who spent all of high school emulating Mahomes.
McCarthy may not just be of that generation, he may be One Of Them.
This is brutally unfair, of course. McCarthy is a true freshman who’s played one (1) college game, that against a MAC team. Meanwhile, Michigan’s starting quarterback had a brilliant day of his own. Cade McNamara went 9-for-11 for 136 yards and two touchdowns. He topped PFF’s (admittedly star-crossed) list of top passers from the weekend:
More importantly, McNamara looked sharp and in command of the offense. Like a quarterback most college programs would start without a competition. Like a quarterback Michigan has been looking to find for most of the Jim Harbaugh era. Like a quarterback for whom we would’ve given up alarmingly important life moments to have on the field against Ohio State most of the years during the losing streak.
But Cade McNamara cannot make that throw.
Maybe that matters this year, maybe it doesn’t. Maybe, improbably, that still doesn’t matter next year. Maybe McCarthy doesn’t have capital-’I’ It and Michigan ends up closer to the Graham Mertz/Jack Coan end of the spectrum than the Kelly Bryant/Trevor Lawrence dream outcome Orlovsky mentioned on the broadcast.
If you can make that throw, you make that throw. The question now becomes if and when you need it. This is a good problem.
Under the old rules, this wouldn’t be a serious discussion. That’s no longer how the game works. Now, a former quarterback mentions the possibility of a midseason starter swap live on ESPN after the incumbent posts arguably the best game of his life. Bringing up that scenario felt justified.
Michigan didn’t need McCarthy to give Baldwin five yards of space out of nothing against Western Michigan. They might not need the three yards of separation that throw could create against Washington, Michigan State, Penn State, and so on.
The six inches of room that throw could open up against Ohio State, though? I’d be lying if I said I weren’t thinking about it. While it isn’t imminent, Harbaugh will eventually have a decision to make.
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