 # Outfield Trigonometry

Baseball as a sport is uniquely obsessed with statistics and measurements. In the case of baseball stadiums, the manner in which the distances to various points in the outfield are measured becomes a crucial issue. Join us as we explore some of the ramifications of this obscure, technical aspect of Our National Pastime.

### Trigonometry 101 for baseball fans

#### Simple case

If the outfield fence is straight, and extends all the way to center field, and is perpendicular to the foul line, we have a right triangle, so it is easy to calculate the center field distance by either of two methods:

1) By using the Pythagorean Theorem: a 2 + b 2 = c 2. The sides (a) and (b) are equal by definition in this case, and the hypotenuse (c) is the (unknown) distance to center field. So, if the distance to the foul line is 330 feet, we get:

c = √ ( 2 x (330 x 330))

c = √ ( 2 x 108,900)

c = √ 217,800

c = 466.690

Note that this matches the pre-1956 dimensions of Shibe Park almost exactly. ( √ is the symbol for square root.)

2) By using the formula below, derived from Trigonometry. The distance (c) to center field can be computed from the foul line distance (a) and the angle (y) from the foul line to the point in question. For center field, that angle is by definition 45°. c = sec (y) ⋅ a

The dot ( ⋅ ) is the algebraic multiplier symbol. The secant is the ratio of the hypotenuse (c) to the adjacent side (a), and equals the inverse of the cosine. The cosine of 45 degrees is .7071, so the secant equals 1.414, which when multiplied by the left field dimension of 330 feet yields 466.690.

c = sec (45°) x 330

c = 1.414 x 330

c = 466.690

This is illustrated on the left side of the adjacent diagram. To find the distance to the power alley(d), we replace the value of the angle (y) with 22.5 degrees (half of 45 degrees; see note on power alley definitions below), the secant of which is 1.082:

d = sec(22.5°) x 330

d = 1.082 x 330

d = 357.189

#### General case

To account for situations in which the outfield fence is not perpendicular to the foul line, we need a more general formula to determine the center field distance:

c = (sin(45°) ⋅ a) ⋅ (1 + tan(x - 45°))

For example, as illustrated on the right side of the diagram above (with the letters marked as prime), if the angle (x') of the fence at the foul pole is 80°, and the distance (a) down the foul line is 340 feet, we have:

c' = (.707 x 340) x (1 + 0.700)

c' = 240.414 x 1.700

c' = 408.752

Note that this nearly matches the dimensions of the old Wrigley Field in Los Angeles. To find the distance to the standard power alley (d) in such situations (where angle x does not equal 90 degrees), we need a new, more complicated formula:

d = sin(67.5°) ⋅ a + (cos(67.5°) ⋅ a) ⋅ tan(x - 67.5°)

Using the previous example (and prime notation), we have:

d' = 0.924 x 340 + (0.383 x 340) x 0.240

d' = 314.119 + 31.263

d' = 345.382

This matches the official dimension data perfectly. This power alley formula can also be applied where the angle of the outfield fence is obtuse (over 90 °), as is the case in many ballparks. In none of them does the fence extend straight all the way to center field, however. See the note below about how the power alleys are defined.

* SINE = Opposite over Hypotenuse, COSINE = Adjacent over Hypotenuse, TANGENT = Opposite over Adjacent. SOH - CAH - TOA !

Left or right field Perpendicular fence
(90°)
Acute angle fence
(80°)
Obtuse angle fence
(100°)
Power alleys Center field Power alleys Center field Power alleys
270292382272325315
275298389277331321
280303396282337327
285308403287343333
290314410293349339
295319417298355344
300325424303361350
305330431308367356
310336438313373362
315341445318379368
320346453323385374
325352460328391379
330357467333397385
335363474338403391
340368481343409397
345373488348415403
350379495353421409
355384502358427415
360390509363433420
365395516368439426
370400523373445432
375406530378451438
380411537383457444
385417544388463450
390422552393469455
395428559398475461
400433566403481467
405438573409487473

NOTE: Data in some of the columns above were previously erroneous, and have been corrected as of February 20, 2016.

### List of relevant stadiums

The trigonometric formulas shown above on this page can be applied to the following stadiums. Those marked with the "#" symbol have straight walls or fences all the way from a foul pole (left or right) to center field. For the others, only the power alley distances can be calculated.

### What about curved outfield fences?

Well, sports fans, that would require a solid grasp of Calculus and differential equations, and I'm afraid I'm too rusty on that subject to take on such a task for the present.

### All about power alley measurements

One of the least-well defined concepts in baseball is the "power alley," which lies somewhere between center field and the foul poles. Supposedly, it represents the vector at which most batters could generate the highest bat velocity and therefore hit the ball the furthest. Unfortunately, there is no clear standard in Major League Baseball as far as exactly which angle it should be measured, and this leads to much confusion. This problem was discovered at RFK Stadium in July 2005, as one example. In some stadiums in which the dimensions are unusually small, they marked the power alley distances at a point much closer to center field than normal; one notorious example was the Kingdome in Seattle. This was based on a definition of the power alley as the line passing through the midpoint between third base and second base, or (the same thing) between first base and second base. That may have been appropriate for old-fashioned ballparks such as Shibe Park in which the outfield walls were perpendicular and extended straight all the way to center field, but it is quite inappropriate otherwise, yielding exaggerated measurements. Just to be clear, I define the power alleys as:

the angular midsections between center field and the left and right foul poles, respectively.

In other words, 22.5 degrees from either the foul line or from center field. The trigonometrical diagram above shows this power alley measurement in black, and the alternative measurement (in which the angle is 26.6 degrees from the foul line and 18.4 degrees from center field) in gray. Line segments e and f are both equal to half of line segment b. Using the latter approach, the power alley would be 369 feet, which matches the originally marked distance to left-center field at Citizens Bank Park.

For ballparks with an acute-angle outfield fence, defining the power alley as the midpoint between center field and the foul pole would yield a smaller difference than ballparks with a perpendicular outfield fence. In the diagram above, line segments e' and f' are both equal to half of line segment b'. In the case of Wrigley Field (L.A.), the "alternative" power alley would be about 348 feet, rather than 345 feet.

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