# Geocaching Coordinate Maths: Projection, Distance and Bearing

How to project a waypoint from a bearing and a distance, how to get distance and bearing between two points, and the two mistakes that send people to the wrong field: coordinate format and magnetic declination.

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- **Canonical URL:** https://dothecalculation.com/blog/hobby/geocaching-coordinate-math
- **Category:** Hobbies & Leisure Utilities
- **Author:** Do The Calculation Team
- **Published:** 2026-09-20
- **Reading time:** 17 min read
- **Publisher:** Do The Calculation (https://dothecalculation.com)
- **Methodology:** https://dothecalculation.com/methodology

---

Puzzle and multi-caches keep asking for the same two calculations. Either you have a point, a bearing and a distance and need the point they lead to, or you have two points and need to know how far apart they are and which way one lies from the other. Both are short pieces of spherical trigonometry, and both are easy to get wrong for reasons that have nothing to do with the trigonometry.

> **Two directions, one problem** — Projection goes forwards: one point plus a bearing and a distance gives another point. Distance and bearing goes backwards: two points give the separation and the direction. They are inverses, which is also the easiest way to check your own work — project a point, run the result back, and you should land on the bearing and distance you started with.

## First: get the coordinate format right

This is the single commonest reason a projection lands in the wrong place, and it is not a maths error at all. The same position can be written three ways, and they look similar enough to paste into the wrong box.

**The same point, three formats**
| Format | Written as | Used by |
| --- | --- | --- |
| Degrees, decimal minutes (DDM) | N 51° 28.057′ W 000° 09.720′ | Geocaching.com and most handheld GPS units |
| Degrees, minutes, seconds (DMS) | N 51° 28′ 03.4″ W 000° 09′ 43.2″ | Traditional maps and some puzzle write-ups |
| Decimal degrees (DD) | 51.467612, −0.161999 | Most calculators, including ours, and mapping APIs |

Geocaching listings use degrees and decimal minutes almost exclusively, so that is the format a puzzle will give you and the format your GPS expects back. Converting is arithmetic, but the trap is that 51° 28.057′ and 51.28057° are different places by about twenty-one kilometres, and both look plausible.

**Converting between formats**

```
Decimal degrees = degrees + minutes ÷ 60
Decimal degrees = degrees + minutes ÷ 60 + seconds ÷ 3600
```
- South latitudes and west longitudes are negative in decimal degrees. A dropped minus sign puts you in the wrong hemisphere, which is a mistake that at least announces itself.
- Keep three decimal places on the minutes. One thousandth of a minute of latitude is about 1.85 metres, so rounding to two costs you the precision a cache hide depends on.

## Projecting a waypoint

This is the shape most puzzle caches take: "from the posted coordinates, go 340 metres on a bearing of 072". The formula walks along a great circle from the starting point.

**Direct (projection) problem**

```
δ = d ÷ R
φ₂ = asin( sin φ₁ · cos δ + cos φ₁ · sin δ · cos θ )
λ₂ = λ₁ + atan2( sin θ · sin δ · cos φ₁ , cos δ − sin φ₁ · sin φ₂ )
```
- φ is latitude and λ is longitude, both in radians. θ is the bearing clockwise from north, d is the distance, and R is the Earth's radius, taken as 6,371,000 m.
- δ is the angular distance: how far around the sphere you travelled, in radians.

**Worked: from N 51° 28.000′ W 000° 10.000′, 340 m on a bearing of 072°**
| Step | Value |
| --- | --- |
| Start, as decimal degrees | 51.466667, −0.166667 |
| Angular distance δ | 340 ÷ 6,371,000 = 0.00005337 rad |
| Result, decimal degrees | 51.467612, −0.161999 |
| Result, as the GPS wants it | N 51° 28.057′ W 000° 09.720′ |

Run that answer back through the inverse calculation and you get 340.00 m on a bearing of 072.000°, which is the check worth doing every time.

Tool: [Project a waypoint](https://dothecalculation.com/calculators/geocaching-projector-calculator) — Enter a starting coordinate, a bearing and a distance, and get the projected point in decimal degrees and in the degrees-and-minutes format your GPS takes.

## Distance and bearing between two points

The reverse problem. Given two coordinates, the great-circle distance comes from the haversine formula and the bearing from the forward azimuth.

**Inverse problem**

```
a = sin²(Δφ ÷ 2) + cos φ₁ · cos φ₂ · sin²(Δλ ÷ 2)
d = 2R · asin( √a )
θ = atan2( sin Δλ · cos φ₂ , cos φ₁ · sin φ₂ − sin φ₁ · cos φ₂ · cos Δλ )
```
- θ comes out between −180° and +180°; add 360 and take the remainder to get a compass bearing from 0 to 360.
- This bearing is the initial bearing. On a long great-circle path the bearing changes as you travel, which is why a flight path looks curved on a flat map. At cache distances it does not change measurably.

For example, from 51.4700, −0.1600 to 51.4750, −0.1500 is 888.2 m on an initial bearing of 051.2°.

Tool: [Distance and bearing between two coordinates](https://dothecalculation.com/calculators/geocaching-distance-bearing-calculator) — Great-circle distance and initial bearing between any two points, in metres, feet, kilometres or miles.

## How wrong is a round Earth?

Both formulas treat the Earth as a sphere of radius 6,371,000 m. It is not one — it is an oblate spheroid, flattened at the poles, and GPS works on the WGS-84 ellipsoid. Rather than wave that away, here is the actual size of the error, computed by running the same pairs through Vincenty's method on WGS-84.

**Spherical versus WGS-84 ellipsoid, from 51.47° N**
| Separation | Spherical | WGS-84 | Difference |
| --- | --- | --- | --- |
| 100 m | 99.96 m | 100.02 m | −0.06 m (0.06%) |
| 1 km | 999.98 m | 1,000.54 m | −0.56 m (0.06%) |
| 10 km | 10,000.1 m | 10,005.8 m | −5.7 m (0.06%) |
| 100 km | 99,672 m | 99,989 m | −317 m (0.32%) |

At the distances a puzzle cache actually uses, the spherical model is out by about six centimetres per hundred metres. A consumer GPS receiver is typically good to a few metres under open sky, so the Earth model is nowhere near your largest source of error. Over a hundred kilometres it starts to matter, but you are not walking that to a cache.

## The mistake that actually sends people to the wrong field

Every bearing in these formulas is relative to true north — the geographic pole. A magnetic compass points at the magnetic pole, which is somewhere else, and the angle between them is the magnetic declination. It varies by where you are, and it drifts year on year.

> **Which north does the puzzle mean?** — If the cache owner means true north and you follow a magnetic compass without correcting, you walk off at the wrong angle. Over 340 m a few degrees is a handful of metres and you will probably still find it; over 2 km in a place with a large declination you will be nowhere near. Most GPS units can display either — check which yours is set to before you trust the arrow.

To convert, add an easterly declination to a magnetic bearing to get true, and subtract a westerly one. For your own location, NOAA's National Centers for Environmental Information publish a declination calculator built on the World Magnetic Model, which is the authoritative source and accounts for the drift.

## Practical notes

- Convert to decimal degrees once, do all the maths there, and convert back at the end. Mixing formats mid-calculation is where errors creep in.
- Keep three decimals on the minutes when you write the answer down — one thousandth of a minute of latitude is about 1.85 m.
- Check a projection by reversing it. If the inverse does not return your starting bearing and distance, something is wrong.
- Check which north the puzzle means, and which north your GPS is displaying.
- For a multi-cache with several legs, project each leg from the previous result rather than trying to combine the bearings. Bearings do not add on a sphere.

## What a minute of latitude and longitude is worth on the ground

Every coordinate calculation eventually comes down to converting angles into metres, and the conversion is different for latitude and longitude. On the spherical Earth used throughout this guide, one minute of latitude is always about 1,853 metres, which is why a nautical mile, defined as exactly 1,852 metres, is so close to it. A minute of longitude is that figure multiplied by the cosine of your latitude, so it shrinks steadily as you move away from the equator.

**Ground distance of one minute of longitude at different latitudes**
| Latitude | One minute of longitude | Compared with a minute of latitude |
| --- | --- | --- |
| 0° (equator) | about 1,853 m | The same |
| 30° | about 1,605 m | 87% |
| 45° | about 1,310 m | 71% |
| 51.5° (London) | about 1,154 m | 62% |
| 60° | about 927 m | 50% |

This is the single fact that makes coordinate arithmetic go wrong most often. A change of 0.100 minutes north and 0.100 minutes east does not move you equally in both directions: at London latitudes the east component is only about 62% as long. Any shortcut that treats a minute of longitude like a minute of latitude points you in the wrong direction as well as to the wrong distance.

## How many decimal places you actually need

Coordinates are often copied with far more or far fewer decimal places than the situation needs. Here is what each decimal place of a decimal-degree latitude is worth on the ground.

**Decimal places in decimal degrees of latitude**
| Decimal places | Example | Resolution on the ground |
| --- | --- | --- |
| 1 | 51.5 | about 11 km |
| 2 | 51.47 | about 1.1 km |
| 3 | 51.468 | about 111 m |
| 4 | 51.4676 | about 11 m |
| 5 | 51.46761 | about 1.1 m |
| 6 | 51.467612 | about 11 cm |

Five decimal places of a degree, or three decimal places of a minute, gets you to roughly a metre or two, which already matches the precision of the receiver in your hand. Six decimal places is finer than any consumer GPS can place you, so the sixth digit in a coordinate someone read off a phone is noise rather than information. Keep it in calculations to avoid rounding error building up, but do not treat it as meaningful.

> **How accurate is the GPS in your hand?** — GPS.gov states that GPS-enabled smartphones are "typically accurate to within a 4.9 m (16 ft.) radius under open sky", and that accuracy "worsens near buildings, bridges, and trees". That sets the real floor on any geocaching calculation: once your arithmetic is good to a metre or two, better maths will not find the cache any faster. Under tree cover, expect to search a wider circle than the numbers suggest.

## A shortcut you can do in the field, and when it stops working

Over short distances you can project a waypoint without trigonometry beyond one cosine, by treating the ground as flat. It is worth knowing because it lets you sanity-check a phone app's answer in your head.

**Flat-ground projection for short distances**

```
Change in latitude (minutes) = distance × cos(bearing) ÷ 1,853
Change in longitude (minutes) = distance × sin(bearing) ÷ (1,853 × cos(latitude))
```
- Distance in metres, bearing measured clockwise from true north.
- For bearings between 90° and 270° the latitude change comes out negative, meaning south; for bearings between 180° and 360° the longitude change is negative, meaning west. Add the results to the starting minutes with their signs.

**How far the flat-ground shortcut misses, compared with the great-circle method, on a bearing of 072° from 51.47° N**
| Distance projected | Miss |
| --- | --- |
| 340 m | about 1 cm |
| 1 km | about 11 cm |
| 5 km | about 2.7 m |
| 20 km | about 43 m |

For the few hundred metres a typical puzzle cache asks for, the shortcut is indistinguishable from the proper calculation; its error is hundreds of times smaller than the GPS error. By five kilometres it is starting to matter, and by twenty it is clearly wrong. Use it to check an answer, not to replace the calculator on long projections.

## A multi-leg walk, worked through

Multi-caches chain projections: each stage gives you a bearing and distance from the last point. Here is a three-leg walk from N 51° 28.000′ W 000° 10.000′, with each leg projected from the result of the one before, exactly as the body of this guide recommends.

**Three legs, each projected from the previous point**
| Leg | Distance and bearing | Arrive at |
| --- | --- | --- |
| 1 | 340 m on 072° | N 51° 28.057′ W 000° 09.720′ |
| 2 | 210 m on 155° | N 51° 27.954′ W 000° 09.643′ |
| 3 | 480 m on 300° | N 51° 28.084′ W 000° 10.003′ |

After walking over a kilometre in total, the final point is only about 155 metres from where you started, because the third leg doubles back. That is typical of multi-caches, and it is a good check on your arithmetic: if the legs obviously turn back on themselves and your answer lands a kilometre away, a bearing has been entered wrongly, usually with the digits swapped or with degrees and minutes confused.

Tool: [Project each leg in turn](https://dothecalculation.com/calculators/geocaching-projector-calculator) — Enter the result of one leg as the starting point of the next, and the answer comes back in the degrees-and-minutes format your GPS expects.

## What ignoring declination actually costs

The body of this guide explains magnetic declination. Here is the size of the mistake if you ignore it. If you walk on a magnetic bearing when the puzzle meant true north, you end up off the intended line by an angle equal to the declination, and the miss grows with distance.

**How far off you end up if you ignore declination**
| Declination at your location | Miss after 340 m | Miss after 2 km |
| --- | --- | --- |
| 2° | about 12 m | about 70 m |
| 5° | about 30 m | about 174 m |
| 10° | about 59 m | about 349 m |
| 15° | about 89 m | about 522 m |

Even a small declination produces a miss several times larger than GPS error over a typical puzzle distance. To convert a true bearing into the magnetic bearing to walk on a compass, subtract an easterly declination or add a westerly one. With 10° west declination, a puzzle bearing of 072° true becomes 082° on the compass. Always get the current value for your location from NOAA's calculator; declination changes slowly from year to year and an old map's figure may be out of date.

## One more setting to check: the datum

A coordinate only means something relative to a model of the Earth's shape, called a datum. GPS itself works on WGS 84, and that is what almost all modern devices and mapping services use. Older paper maps and some national grids use other datums, and the same numbers on a different datum can point to a place a noticeable distance away. If a projection lands consistently offset from where it should, check that your device and your source coordinates are on the same datum before suspecting the arithmetic.

## When the numbers are right and the cache is not there

- **Let the receiver settle.** Stand still for a minute at ground zero. A position that has been moving tends to lag behind you.
- **Average.** Take several readings a little apart in time and use the middle. Some devices have a waypoint-averaging function for exactly this.
- **Search a circle, not a point.** Under open sky that circle is roughly five metres in radius by GPS.gov's figure; under trees or near buildings, widen it.
- **Re-check the input rather than the maths.** The common faults are a swapped digit, degrees and minutes confused, a missing minus sign for west or south, and a magnetic bearing treated as true.
- **Project back.** Run your answer through the distance and bearing calculator from the start point. If it does not return the bearing and distance you were given, the error is in what you typed.

## Three format conversions, worked

Converting between formats is where most copying errors creep in, so it is worth doing a few by hand once. Each conversion below was checked by running it back the other way.

**Converting between the three coordinate formats**
| Starting from | Working | Result |
| --- | --- | --- |
| N 51° 28′ 03.4″ (DMS) | 51 + 28 ÷ 60 + 3.4 ÷ 3600 | 51.467611 in decimal degrees, or N 51° 28.057′ in degrees and decimal minutes |
| W 000° 09′ 43.2″ (DMS) | 0 + 9 ÷ 60 + 43.2 ÷ 3600, then negative because it is west | −0.162000, or W 000° 09.720′ |
| N 40° 26.767′ (DDM) | 40 + 26.767 ÷ 60 | 40.446117 in decimal degrees; the 0.767 minutes is 46.0 seconds, so N 40° 26′ 46.0″ in DMS |
| −33.856784 (DD) | 33 whole degrees, then 0.856784 × 60 minutes, south because negative | S 33° 51.407′ |
| 151.215297 (DD) | 151 whole degrees, then 0.215297 × 60 minutes, east because positive | E 151° 12.918′ |

Two habits prevent almost every conversion error. Carry the hemisphere letter or the sign through every step instead of adding it at the end, and keep the minutes to three decimal places, which is what geocaching listings use. If a converted coordinate ever shows minutes of 60 or more, something has gone wrong: 60 minutes is a whole degree and should have been carried.

## True north, magnetic north and grid north

Declination is the difference between true north and magnetic north, but there is a third north that catches people who navigate with paper maps. National mapping grids, such as the ones printed on many hiking maps, have vertical grid lines that point to grid north. Because a flat grid is laid over a curved Earth, grid north and true north differ slightly over most of the map, and the difference changes across it.

- **True north** is the direction of the geographic pole. The formulas in this guide, and geocaching puzzles unless they say otherwise, use it.
- **Magnetic north** is where a compass needle points. The angle between it and true north is the magnetic declination, which NOAA's calculator gives for your location and date.
- **Grid north** is the direction of the vertical grid lines on a projected map. Maps that use it usually print the relationship between all three norths in the margin.

The practical rule is to decide which north each number refers to before you do anything with it. A bearing measured on a paper map with a protractor is a grid bearing; a bearing from a compass is magnetic; a bearing from this guide's formulas or a puzzle is true unless stated. Mixing them is how an otherwise perfect calculation walks you into the wrong field. Where a map margin gives the angles, use those figures for that map rather than a general rule.

## Sources

- The projection and haversine relations are the standard great-circle formulas on a sphere; the Earth radius of 6,371,000 m is the conventional mean radius.
- The spherical-versus-ellipsoid comparison above was computed for this guide by running the same coordinate pairs through Vincenty's inverse method on the WGS-84 ellipsoid (a = 6,378,137 m, f = 1/298.257223563). It is a derivation, not a quoted figure.
- NOAA National Centers for Environmental Information, magnetic declination calculator, built on the World Magnetic Model: https://www.ngdc.noaa.gov/geomag/calculators/magcalc.shtml

## Where to go next

Once you know where you are going, the other question is how long it will take. [How long will this hike take, and how heavy should my pack be](/blog/hobby/hiking-time-and-pack-weight) covers route timing and load, and both bear on the same decision about whether a leg is walkable in the daylight you have.

## Common questions

**How do I project a waypoint from a bearing and distance?**

Convert the start point to decimal degrees, divide the distance by the Earth radius to get an angular distance, then apply the great-circle direct formula. From N 51° 28.000′ W 000° 10.000′, going 340 m on a bearing of 072° lands on N 51° 28.057′ W 000° 09.720′.

**Why did my projected coordinate land in the wrong place?**

Nine times out of ten it is the coordinate format rather than the maths. Geocaching uses degrees and decimal minutes; most calculators use decimal degrees. Treating 51° 28.057′ as 51.28057° puts you about twenty-one kilometres away. The other common cause is following a magnetic bearing when the puzzle meant true north.

**Is the haversine formula accurate enough for geocaching?**

Comfortably. Against the WGS-84 ellipsoid it is out by about 0.06% at short range — six centimetres per hundred metres — while a consumer GPS is typically good to a few metres. The Earth model is not your limiting factor.

**What is magnetic declination and do I need it?**

The angle between true north and magnetic north at your location, which also drifts over time. You need it whenever a bearing from a calculation is followed on a magnetic compass. Add an easterly declination to convert magnetic to true, subtract a westerly one. NOAA publishes a calculator for your location.

**What is an initial bearing?**

The direction to set off in along a great circle. On a long path the bearing changes as you travel, which is why great-circle routes look curved on a flat map. Over the distances in a cache puzzle the change is not measurable.

**Can I add two bearings together for a multi-leg walk?**

No. Bearings do not add on a sphere. Project the first leg, then project the second leg from the result of the first, and so on. Each leg starts from the previous answer.

**How many metres is one minute of latitude?**

About 1,853 metres on the spherical Earth used for these calculations, which is why a nautical mile, defined as 1,852 metres, is so close to it. A minute of longitude is shorter away from the equator: about 1,154 metres at London's latitude and about 927 metres at 60° north.

**How accurate is a phone GPS for geocaching?**

GPS.gov states that GPS-enabled smartphones are typically accurate to within a 4.9 metre radius under open sky, and that accuracy worsens near buildings, bridges and trees. That means your arithmetic only needs to be good to a metre or two; past that, searching carefully matters more than better maths.

**Can I work out a projection without a calculator?**

Over short distances, yes. Change in latitude in minutes is the distance times the cosine of the bearing, divided by 1,853. Change in longitude is the distance times the sine of the bearing, divided by 1,853 times the cosine of your latitude. Over 340 metres it agrees with the great-circle method to about a centimetre; by 20 km it is off by over 40 metres.

**How many decimal places should a coordinate have?**

Five decimal places of a degree, or three decimal places of a minute, gives about one to two metres, matching consumer GPS precision. A sixth decimal place of a degree is about 11 centimetres, finer than your receiver can place you.

**How far off will I be if I ignore magnetic declination?**

It depends on the declination and the distance. With 5° of declination you miss by about 30 metres over 340 metres and about 174 metres over 2 kilometres. With 10° those become about 59 and 349 metres.

**How do I convert degrees, minutes and seconds to decimal minutes?**

Keep the degrees, and divide the seconds by 60 to add to the minutes. N 51° 28′ 03.4″ becomes 28 + 3.4 ÷ 60 = 28.057 minutes, so N 51° 28.057′. To go all the way to decimal degrees, divide the minutes by 60 and add them to the degrees: 51.467611.

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_Source: [Do The Calculation](https://dothecalculation.com/blog/hobby/geocaching-coordinate-math). Quote freely with attribution and a link to this page._