A Visualized Primer on Vertical Approach Angle (VAA)

This time last year, I investigated where vertical approach angle (VAA) seems to matter most. The short answer: at the top of the strike zone for four-seam fastballs and at the bottom of the zone for sinkers and two-seam fastballs. This piece, which is adapted from a presentation I did as part of the 2022 PitcherList PitchCon, will provide much-needed additional context, like benchmarking and watermelon-colored heat-map-style graphics.

For the uninitiated — which could be many of you — VAA is the angle at which a pitch approaches home plate… vertically. Despite its usefulness, the concept has experienced slow uptake in the public sphere. I think that’s largely due to a lack of data, which, for nerds like me too entrenched in baseball Twitter, has shrouded the metric in mystery. Why are scouts and college baseball R&D departments valuing VAA so highly, why have I barely heard of it, and how can I find it?

To answer the last question: Statcast is granular enough that, fortunately, we can calculate VAA using physics. So, let’s calculate it! Thanks to Baseball Prospectus‘ Harry Pavlidis (who credits baseball’s renowned physicists), here are the equations:

vy_f = -sqrt(vy0² – (2 * ay * (y0yf)))
t = (vy_fvy0) / ay
vz_f = vz0 + (az * t)
VAA = -arctan(vz_f/vy_f) * (180 / pi)

where, per Statcast’s documentation:

  • vy0 = The velocity of the pitch, in feet per second, in y-dimension*, determined at y=50 feet. (*toward home plate)
  • ay = The acceleration of the pitch, in feet per second per second, in y-dimension, determined at y=50 feet.
  • y0 = 50 (“y=50 feet”).
  • yf = 17/12 (home plate, converted to inches).
  • vz0 = The velocity of the pitch, in feet per second, in z-dimension**, determined at y=50 feet. (**vertically)
  • az = The acceleration of the pitch, in feet per second per second, in z-dimension, determined at y=50 feet.

Let’s work on that public uptake!

Now it’s primer time.

VAA, being a function of velocity and acceleration in all three dimensions, is largely a product of these physical specifications:

  • Release speed (a.k.a. velocity),
  • Pitch height, and
  • Release point.

VAA correlates strongly with all three, and changes to any of these specs will naturally alter VAA. Here, we can see just how strongly VAA (x-axis) correlates with pitch height (y-axis):

Generally, this amorphous blob illustrates two points: (1) certain VAAs can only be achieved at certain pitch heights, and (2) all VAAs can be achieved at many pitch heights. If all we know about a pitch is that its VAA is –5.0°, we can intuit that the pitch was thrown anywhere between the bottom of the strike zone and a half foot above it — not to mention, we would know little about the velocity or the release point. In a vacuum, that VAA is not particularly helpful. It lacks context.

So let’s add some context! We can convert raw VAA measurements into “VAA Above Average” (VAAAA? Please, Alex, no.) by normalizing for pitch height. If we adjust for pitch height among all pitches at once, with four-seamers, sinkers, curves and so on all compared to one another, our amorphous blob is, well, still amorphous. But it is visibly less correlated with pitch height:

And if we adjust for pitch height within a specific pitch type — like for four-seamers, as you’ll see below — our VAA Above Average blob becomes a nearly perfect circle:

The same is true with change-ups (the line above the graph is the top of the strike zone):

Having added height-related context, if I tell you a pitcher’s four-seamer’s VAA is 0.8° above average, you can know two things immediately: (1) his four-seamer is significantly flatter than average, irrespective of pitch height; and (2) he has (i) above-average velocity and/or (ii) a lower-than-average release point. Flip it to 0.8° below average, and all the inverse points similarly apply.

Moreover, comparing within a pitch type (rather than including all pitch types) creates a uniform, normally distributed scale:

VAA Distribution by Pitch Type
+/- StDev Four-Seamer Sinker Cutter Change-Up Slider Curve
-3 σ -1.5° -1.5° -1.6° -1.6° -2.1° -2.4°
-2 σ -1.0° -1.1° -1.3° -1.1° -1.5° -1.9°
-1 σ -0.5° -0.6° -0.7° -0.6° -0.8° -1.1°
0 σ 0.0° 0.0° 0.0° 0.0° 0.0° 0.0°
+1 σ +0.5° +0.5° +0.6° +0.5° +0.7° +1.0°
+2 σ +0.9° +1.2° +1.1° +1.0° +1.4° +1.9°
+3 σ +1.4° +1.8° +1.4° +1.4° +1.9° +2.4°
SOURCE: Statcast

Here it is converted to 20-80 scouting grades:

VAA Scouting Grades by Pitch Type
Scouting Grade Four-Seamer Sinker Cutter Change-Up Slider Curve
20 -1.5° +1.8° n/a n/a n/a n/a
30 -1.0° +1.2° n/a n/a n/a n/a
40 -0.5° +0.5° n/a n/a n/a n/a
50 0.0° 0.0° n/a n/a n/a n/a
60 +0.5° -0.6° n/a n/a n/a n/a
70 +0.9° -1.1° n/a n/a n/a n/a
80 +1.4° -1.5° n/a n/a n/a n/a
SOURCE: Statcast

The stark contrast between these tables is important. Generally, a pitch’s VAA is only as important as its velocity; the higher the velocity, the larger the role VAA might play in its success, and vice versa. Thus, breaking and off-speed pitches, universally thrown at lower velocities than their counterpart fastballs, rely minimally on VAA to succeed. Let’s dive a little deeper into some individual pitch types.


The flatter a pitcher’s four-seam fastball, (1) the lower in the zone it can induce whiffs (per swing), and (2) the larger its margin for error throughout the zone. A flat four-seamer can find whiffs throughout the strike zone, especially at the top:

Whiff rate (whiffs per swing, above) is only as valuable as the swing rate (Swing%) accompanying it; a high whiff rate with a low swing rate doesn’t mean much. In addition to generating more whiffs up, flatter four-seamers also induce more swings up…

… which, when combined with whiffs, establish a swinging strike rate (SwStr%) sweet spot at the top of the zone, where the majors’ best pitchers thrive. That sweet spot exists for steep (or even average) four-seamers, too, but with a much smaller margin for error:

(An aside, because flat four-seamers tend to lure hitters’ eyes up, flat four-seamers can also get away with chasing called strikes low in the zone. Walker Buehler fancies himself a nice, low called strike every now and then, and it helps explain why some pitchers are better at getting hitters to leave their bats on their shoulders in critical counts.)

To capture overall value, I developed Deserved ERA (dERA), which uses “deserved” (a.k.a. regressed) versions of strikeout and walk rates (K% and BB%) and, to measure contact quality, Statcast’s expected wOBA on contact (xwOBAcon) metric. Because dERA accounts for, and thus is calculated for, every pitch, we can use dERA like a measure of pitch quality on an ERA scale and use it to create a value-based heat map:

Unsurprisingly, four-seam effectiveness tracks with swinging strike efficacy. However, there’s one skill factor embedded in dERA not visualized above: pop ups. In embracing the illusion of flatter fastballs “rising” up over a swinging bat, a hitter is more likely to hit a pop up at the top of the zone and, compared to a steeper four-seamer, more likely to produce a pop up in more vulnerable parts of the strike zone:

Aces leverage this lethal combination of whiffs and weak contact at the top of the zone. Because of his velocity, Jacob deGrom’s fastball is so flat and punishing that he can jam it down the heart of the zone, possibly without a care in the world. That he would even think to venture north with it should be penalized by law. Plus velocity isn’t everything, however; bad location and suboptimal arm slots can undercut it, margin for error be damned.

Jacob deGrom is an established ace. So, too, is Shane Bieber, but he can be prone to bouts of hard contact (evidently I’m on an island when it comes to caring about this). Meanwhile, Germán Márquez tantalizes us with two excellent secondaries, while his elusive success is attributed to having to call the dreaded Coors Field home. What can VAA tell us about the quality of their four-seamers and how it relates to their success overall?

A whole heck of a lot, pal. From left to right, here are approximations of Márquez’s, Bieber’s, and deGrom’s four-seamers overlaid on our handy dandy watermelon four-seamer dERA heat map. The shaded portions are my best attempt to indicate saturations of higher pitch frequency:

deGrom’s four-seamer finds a lot of green, Bieber’s finds some green but is largely concentrated in a sea of red, and Márquez’s has hardly any green to speak of. Comparing four-seamer dERA to all pitches puts into perspective how damaging a bad four-seamer (or fastball in general) can be. Márquez’s lack of success isn’t necessarily a product of Coors, and Bieber’s hard contact issues aren’t necessarily fluky:

Sinkers (and Two-Seamers)

Sinkers, like four-seamers, are sensitive to VAA — just a little bit less so, and in different ways. We see the same trends in whiffs per swing, just without the increasing margin for error…

… as well as the same upward trend in swing rate, but even more pronounced…

… resulting in similar pockets of swinging strike effectiveness:

Like flat four-seamers up, it’s down in the zone where steep sinkers create extreme launch angles (in this case, weak grounders) that, when combined with above-average swinging strike rates, create a sweet spot for success. And, like their four-seam counterparts, steeper sinkers have a wider margin for error throughout the zone for inducing this weak contact:

VAA results for cutters aren’t particularly interesting — they, too, aren’t all that sensitive to VAA — so I am choosing to exclude them for brevity.

Breaking and Off-Speed Stuff

As I mentioned, VAA exerts less influence on the effectiveness of breaking and off-speed pitches. Change-up whiff rates are generally agnostic to VAA, inducing more whiffs down (and fewer whiffs up) regardless of the VAA Above Average:

The same is true for sliders and curves, each with larger and smaller bands of effectiveness (represented by the rectangles), respectively, but the same indifference to VAA Above Average:

The remaining graphics I omitted look similar, too: VAA Above Average simply doesn’t move the needle much for whiff rates, swing rates, dERA, or anything else. That doesn’t mean VAA doesn’t matter for these types of offerings — for some pitchers, it must. It’s just that, by and large, these offerings exhibit an insensitivity to VAA, relying more on other characteristics (like location) to succeed.

Bringing It All Back Together

With dERA colorized on a uniform scale for all pitches, we can see where each pitch type succeeds or fails. Graphic design is my passion:

In this house, we celebrate flat four-seamers up (with gutsy, elite four-seamers down the middle), steep sinkers down, all kinds of breaking and off-speed pitches down, and a hanging secondary in favor of a hanging fastball. Also, I know the heat map makes it looks enticing, but I do not endorse spiking pitches in the dirt. Sorry.

Final Thoughts

VAA can be many things, but it isn’t everything. I am reluctant to overstate its importance, although I am also reluctant to understate it. This analysis is deliberately reductive to illustrate broader points. I have made a loose declaration as to where flat four-seamers and steep sinkers work best and why (they appear to rise over/dive under the bat), but they don’t always work for the same reasons — it could be velocity, it could be release point, it could be those dastardly seam-shifted wake effects. Realistically, it’s some combination of all these things, interacting slightly differently for each pitch and pitcher.

VAA is useful, though, because it captures the interaction of multiple pitch attributes at once. Sometimes, we don’t know exactly which attribute within VAA is wielding the greatest influence on a pitcher’s success: deGrom thrives via velocity, whereas Paul Sewald found breakout success thanks to a lower arm slot that begets an absurdly flat fastball. VAA Above Average is even more useful because it helps us cut through this noise a little more quickly. And to be clear, you can also adjust VAA in a myriad of ways. I chose a pitch height adjustment because of the launch angle implications, which are important to contact quality. But it’s not the only way!

The cultivation of a flat four-seamer or a steep sinker is the cultivation of capital-S Stuff. Stuff buys you wider margins for error; it’s why some pitchers can afford to just throw it down the middle. But I will argue, and stubbornly expect you to agree, that many pitchers lack the requisite Stuff to throw it down the middle, with such a decision ranging from “fine, but not ideal” to “extremely ill-advised.”

Accordingly, the less impressive a pitcher’s Stuff, the more dependent he probably ought to be on pitch location. But! You can still achieve good outcomes with bad pitch shapes. Let us not resign pitchers to predetermined fates. Some pitchers will defy the expectations set here. It should be noted, however, that those pitchers are probably outliers in their own right, using extreme pitch specs in atypical ways. This author is convinced there isn’t a “best” way, just different ways, specifically ways that emphasize uniqueness and separate oneself in a manner hitters don’t usually encounter. An elite VAA (Above Average) kind of serves as a nice shorthand for this concept.

It occurs to me that pitchers who (1) throw two fastballs, like a four-seamer and a sinker; (2) can mirror the release speeds, release points, and spin axes of the two fastballs; and (3) can generate wildly different movement profiles on each (especially if the movement includes optimal VAAs in optimal locations) can succeed wonderfully. That’s Zack Wheeler, and Lance Lynn, and Pablo López, just to name a few — and they’re all pretty dang good. I could write an entire post about pitchers who do this (don’t make me!). Ben Clemens happened to get that conversation started this past Friday, highlighting Wheeler, among others, as someone who weaponizes his fastballs to exploit hitter fly ball/groundball splits. I digress, but this thread has barely been pulled.

Lastly, on my Pitch Leaderboard, I host data for VAA (‘Specs’ tab) and VAA Above Average (‘Experimental Stats’ tab). If you’re a baseball nerd who is interested in VAA but wise enough to not want to deal with millions of lines of Statcast data, I hope the Pitch Leaderboard can help you!

Currently investigating the relationship between pitcher effectiveness and beard density. Two-time FSWA award winner, including 2018 Baseball Writer of the Year, and 8-time award finalist. Featured in Lindy's magazine (2018, 2019), Rotowire magazine (2021), and Baseball Prospectus (2022). Tout Wars competitor. Biased toward a nicely rolled baseball pant.

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6 months ago

This is outstanding! Thank you.