North Dakota - South Dakota boundary marker calculations

ND/SD boundary marker #276

The quartzite monuments along the North Dakota - South Dakota USA boundary are rich in history. But we'll leave that for the historians. Let's calculate their positions!


Given a monument's mile number, calculate its location

In the above photograph we see monument (mile) number 276. Where was it taken?
North Dakota-wide Burkle addressing system addresses:

9201 102 ST SW

Burkle intersections:

92 AVE & 102 ST SW

In fact because the Burkle system is so grand, I made some maps of it.

North Dakota PLSS township and range system:

SE corner of Section 34 T129N R93W

Latitude and longitude:

Sorry, the boundary (7th Standard Parallel) isn't as straight as it seems. So no simple exact formula is available. The KMLs I tried to make looked embarrassing.

Lake Traverse Indian Reservation

And sorry, all bets are off from mile 12 to 44.5 as the boundary crosses the Lake Traverse Indian Reservation (Sisseton Indian Reservation on maps), with its different grid plus story behind it. The markers however still connect to the PLSS if one extrapolates north out of the Reservation.

Given a location, calculate the mile number

If there was a boundary marker where US highway 85 crosses the ND/SD state line, which is at

what would be its number? Answer: Mile 329.


Topographic maps

Topographic maps (available on the USGS website) sometimes show every marker's position, numbered too, that existed at the time of publication. And of course they also show the PLSS grid.

Geodetic points

National Geodetic Survey data sheets online say e.g.,



When tagging, how monumental are the monuments?

Bar marks

We notice the two bar "=" mark on the monument in the photograph. We read elsewhere that that meant "two miles from the township corner." Likewise for 一, 二, 三...

And indeed our photo, at the southeast corner of section 34, is two miles from the southeast corner of section 36, which is the southeast corner of a US township. Ah, so, at least here, the "=" refers to the previous (southeast) township corner, not the next (southwest) corner.

Origin point

So where's the system's "origin"? We extrapolate: inputting 0, getting...

...which blasts us eastward, beyond the famous "Initial Point 9 Chains East" marker, and lands us at a mundane section corner in Minnesota (same principal meridian.)


Just have a look in our source code.

Yes it's "tough luck" that we must give

... 151 _ 101 _ 51 _ 1 W|E _ 49 _ 99 _ 149 _ 199 ...

due to quadrants, but we always want to discuss the southeast corners of sections. (The above diagram is a close up of virtual house numbers for miles 186 through 182.)

(Alas, that's the problem with quadrants. One needs four templates for local officials to use, instead of just one. South Dakota is a breath of fresh air, doing away with quadrants. (If one of their axis started at 500 miles (600 even fairer, each axis has 400 to grow) instead of 100, then it's OK even if a 911 caller forgets avenue vs. street. More other house numbering discussions.))

Using South Dakota systems

At first look, the boundary markers are tied to the North Dakota side's PLSS, and thus Burkle addressing system. South Dakota roads don't "hit" them as exactly.

Let's take another look at South Dakota's avenue numbers. Maybe they'll be useful after all, especially in the Lake Traverse Indian Reservation, say following the R51E/R50E the North Dakota side jumps by about 1 + 5/8 miles, but the South Dakota side remains stable.

(By the way, jumping will create an address gap at one end, and much worse: duplicate addresses at the other. Not a problem now, but one day when the land fills up with houses... So how did they deal with it? Well that's beyond our scope today.)

East end

The first four corner highway intersection is at Monument mile 1, indeed nicely at predicted North Dakota 183rd Avenue SE. On the South Dakota side is their 480th Avenue. 480 = 183 + 297. SD = ND + 297... Not so fast! We note that the SD avenues extend from the previous correction line, (hmm, at R58W I had to look nine townships (6 × 9 = 54 miles) south to find a jog...) and converge due to earth curvature. They don't stack up to the North Dakota ones, mile per mile. They're shorter. We need another data point...

Oh and by the way, it turns out South Dakota's avenue numbers are hard to predict. One would think that, "Oh, they just skipped a few here and there due to earth curvature." However there is even some skipping way down along the Nebraska border.

West end

Our Terminal Monument, at the Montana / North Dakota / South Dakota tri-point, is at 360 miles + 45.35 chains = 360.566875 miles.

(Wait, I can't resist. Where's that? Answer: 176P AVE & 102 ST SW; 17657 102 ST SW; 0.566875 mi. w of SE cr S34 T129N R107W; (Yes, "176P" Avenue.) (Also remember, most addresses in this article are merely virtual.))

South Dakota Avenue 100, the virtual avenue that runs north and south along the Wyoming / South Dakota state line, hits an (interesting, read all about it elsewhere) jog at the Wyoming / Montana / South Dakota tri-point, and disappears. More stories.

Well, OK. On a topographic map we measure that our Terminal Monument is 5 miles + 1989 ft = 5.3767 miles west of the place where South Dakota's Black Hills Meridian system's R1E / R2E line ("Avenue 106") hits the North Dakota state line.

Fine. So our Terminal Monument is at South Dakota avenue 106 - 5.3767 = 100.6233.

OK, so we have traveled westward 360.566875 - 1 = 359.566875 monument miles (same as North Dakota avenue miles.) And in doing so we have also gone 100.6233 - 480 = -379.3767 South Dakota avenues.


OK, I made formulas for "Given Monument mile number, compute South Dakota Avenue number", and the inverse, and put them in the aforementioned source code.

Adams County North Dakota system

Adams County North Dakota has their own road numbering system, different than the state-wide Burkle system. It has a twist: Adams: 3 2 1 2 3; Burkle: 2 1 0 2 1. (Miles surrounding center axis.) (Also Adams' 1 ST makes a real twist, half a mile to the west, just before it hitting the state line - surely to politely line itself up with the South Dakota grid before entering that state.) Anyway, we also put it in our source code.


Wouldn't it be more useful to the general public to just make programs to convert between all the other systems mentioned on this page, and never mind the dusty monuments? Well, yes, but, well, it's time for bed. Besides, others have surely done that already.

Manz, L.A., 2013, The Public Land Survey System (PLSS) – Part 1


In fact using BLM PLSS data, I made a .KMZ file of predicted monument locations! Looks great viewed over a topographic map with that's red section lines. I used the "Viking GPS Data and Topo Analyzer..." to see it.

As far as the part through the Reservation, well those PLSS sections don't have corners on the boundary monuments, and extrapolating southward would lead to sloppy results.

Also I threw away odd shaped sections' troublemaker nodes. And e.g., the Initial Point isn't part of PLSS data either.

Dan Jacobson

Last modified: 2023-12-28 22:30:57 UTC