How to Triangulate a City From Compass Bearings

Triangulating a position from compass bearings is a navigation method that’s been used to find both hikers on a hillside and cities on a map for as long as compasses and maps have existed together. There are two widely used terms in triangulation, namely resection and intersection. Learn them and a game like WorldleCity’s GeoAngles stops feeling like guesswork and starts feeling like a puzzle you can reason through. Of course, playing GeoAngles is not the only reason you should learn them, as triangulation can be a real life-saving skill.

---

Resection vs. intersection: the two directions of the same idea

Resection and intersection deal with the same thing, the compass bearing, but point it in opposite directions.

• Resection — Imagine you are a hiker-with-a-compass and you want to figure out your own position on a map. You take bearings, using your compass, out to two or three landmarks you can identify on the map. You then draw a line from each landmark back along that bearing. Where the lines cross on the map is your position.
• Intersection — Same setup as above, but now you want to find the position of a target on the map. You stand at two or more known points, take a bearing toward the target from each, and draw a line from your position outward to the target. Where those lines cross on the map is the target’s position.

Resection finds your own position from known points. Intersection finds an unknown point from your own known positions. They’re mirror images of each other, and converting between them is just a matter of flipping each bearing 180 degrees, known as taking its back bearing. A landmark that reads 40° from where you stand reads 220° when taken from the landmark back toward you.

How to actually do it, step by step

The Ordnance Survey’s field method is a good real-world walkthrough, built for hikers doing resection with a map and compass. The steps transfer directly to reasoning about any triangulation problem:

1. Pick landmarks spread around you, not clustered together. Three features spaced roughly 120° apart (one ahead and one over each shoulder) are far more reliable than three in roughly the same direction.
2. Take each bearing carefully. Point the compass at the landmark and read the angle off the dial. Avoid parallax errors while taking the readings.
3. Correct for magnetic variation if needed. A magnetic compass points at magnetic north, not true north. The gap between the two (magnetic declination) differs by location and drifts slowly over time. Declination is positive when magnetic north lies east of true north and negative when it lies west, so a true bearing is the magnetic bearing plus the declination (NOAA’s World Magnetic Model publishes current values). Apply the correction before plotting on a map that’s drawn to true or grid north.
4. Reverse each bearing, then draw the line. Add or subtract 180° and draw that back bearing on the map from each landmark as a full line.
5. Read the triangle, not just the crossing point. With three lines, you’ll rarely get one perfect intersection. Most often you’ll get a small triangle. The larger the triangle, the less you should trust the fix.

That small triangle of error is called the cocked hat, an informal name among navigators, after the three-cornered hats it resembles. It isn’t a flaw in the method; it’s the method telling you how confident to be.

The bearing math underneath it

Real navigation over long distances uses the initial bearing or forward azimuth, computed with the great-circle formulas that Movable Type’s reference implementation documents in detail. GeoAngles uses a simpler flat-map approximation instead: it scales the east-west difference by the cosine of the mean latitude and takes the angle of that, which lands very close to the heading you would read off a Mercator map. That is why its arrows can be reversed and drawn as straight lines on a Mercator map with only small error over short distances. If you want to see why straight lines on a map and shortest routes are different things, the Mercator projection explained covers it.

Triangulating a city with GeoAngles

GeoAngles is a practical, low-stakes way to run this exact skill without a compass or a hillside. The board shows four known cities, each with an arrow pointing outward from a hidden mystery city. That’s resection geometry: the same bearings a hiker would take from an unknown position, read as if you were standing in the mystery city.

The reason GeoAngles clicks is that each arrow you see is a bearing from the mystery city to a known one. To use it the way a navigator would, flip it into its back bearing, the direction from the known city back toward the mystery one, and imagine that line drawn on a world map. Do that for two or three of the four known cities and you’ve done the plotting step of resection in your head. Where your mental lines would cross is roughly where the mystery city sits.

You get six guesses, and the board itself gets more precise with each guess. The mystery city starts fully hidden, then resolves through a distance-band color, then a rounded-to-500-km ring, then the exact distance, so even a rough first triangulation gets corrected round by round. You pick your answer from a searchable city list, so once you’ve narrowed the region, spelling issues never stand between you and the right guess. Try today’s board at worldlecity.com/geoangles, or practice the skill at your own pace, with a selectable difficulty, at worldlecity.com/geoangles/practice.

A real board, solved

GeoAngles Quiz #9, played on 11 September 2026, showed arrows from the hidden city to Freetown (310.5°), Santiago (264.0°), Cape Town (215.1°) and Islamabad (38.9°). Of course, the naked eye can’t judge angles this precisely, but with practice you can estimate the range. Reversed, they become 130.5°, 84.0°, 35.1° and 218.9°. Drawn from each clue city, the four lines look like this:

Three of the four lines pass within 40 km of Gaborone, the answer. The cocked hat formed here is small, so a guess can be made with confidence. However, Islamabad’s line misses by 174 km, a gap that comes from GeoAngles using a flat-map approximation rather than exact great-circle bearings. Factoring in Islamabad’s line may throw off your guess, which adds to the level’s difficulty.

How accurate is this, really?

I simulated 825 GeoAngles boards, five per city across 165 random cities in the game’s pool, using the game’s own clue picker. For each board I reversed the arrows, found where the lines cross and measured how far that point landed from the true city.

Arrows used	Median error	90th percentile	Within 100 km
2	55 km	413 km	62%
3	51 km	233 km	67%
4	54 km	180 km	68%

I found that the typical error is roughly the same regardless of how many clues are used. Extra arrows don’t shrink the typical error, which stays around 50 km. However, they shrink the worst case. With all four arrows, no board missed by more than 500 km. The size of the cocked hat also tells you how much to trust the result. On boards where the four lines clustered tightly, the median error was 14 km. Where they spread widely, it was 132 km.

These numbers assume every arrow is read perfectly. A person judging angles by eye will do worse, so treat them as the limit of what the clues allow, not as a promise of what you will achieve.

Why the skill is worth having beyond the game

Resection and intersection aren’t retired techniques. They’re still the backbone of land and inshore navigation whenever GPS isn’t available or isn’t trusted, precisely because the method only needs a map and an inexpensive compass to work. Learning to read a bearing as a line rather than a single number, and to spot when three lines agree with each other versus when they’re telling you to distrust the fix, is a genuinely transferable piece of spatial reasoning. A game board is just a faster, lower-stakes place to build the instinct before you ever need it on a hillside.

Sources

• Position resection and intersection — Wikipedia
• Pinpointing your location with bearings and triangulation — Ordnance Survey
• Calculate distance and bearing between two latitude/longitude points — Movable Type Scripts
• World Magnetic Model — NOAA National Centers for Environmental Information

FAQ

What’s the difference between resection and intersection?

Resection is when you stand at an unknown point and take bearings out to known landmarks to find your own position. Intersection is the reverse: you stand at known points and take bearings toward a single unknown target to find its position.

Why do you need three bearings instead of two?

Two bearings will already give you an approximate fix where the lines cross, but any small error in either bearing shifts that crossing point without warning you it happened. A third bearing usually misses the same point by a small amount, and that gap, called a cocked hat, tells you how much to trust the fix.

What is a back bearing?

A back bearing is a compass bearing rotated 180 degrees. If a landmark reads 40 degrees from where you’re standing, the back bearing, the direction from the landmark to you, is 220 degrees. Converting between the two is what lets you switch between resection and intersection using the same set of angles.

How does GeoAngles turn this into a game?

GeoAngles shows you the bearing from a hidden mystery city outward to four known cities, which is the resection geometry already computed for you. To find the mystery city, you mentally reverse each arrow into a back bearing running from a known city toward the hidden one, the same reversal a hiker does by hand, then guess where those reversed lines would converge.

How accurate is triangulating a city from GeoAngles arrows?

With perfectly read arrows, a simulation of 825 boards put the crossing point a median 50 km or so from the answer, and more arrows mainly removed the worst misses: the 90th percentile fell from 413 km with two arrows to 180 km with four. A person reading angles by eye will be less precise, so the figures are a ceiling, not a promise.

Recommended

• Proximity Feedback in Games: Read Your Hints Like a Pro
• What Is the Mercator Projection? The Map That Gets Direction Right
• Estimate Map Distance in 10 Seconds
• What Is a Daily City Challenge? Rules and How to Play