---
title: "Destination Point from Bearing and Distance | North Pinnacle"
description: "Find where a bearing and a distance take you from any start point, on the WGS84 ellipsoid and on a sphere, with the final bearing and an MGRS reference."
url: https://northpinnacle.top/destination-point/
---

Sheet 4 · Destination point

# Destination point from a bearing and a distance

Start somewhere, set off on a bearing, travel a distance: here is where you end up and which way you'll be facing.

## Readout

Destination, 10.000 km on 045.0°

56.860345, -4.887680

On a sphere of mean radius the same trip ends at 56.86044, -4.88728, 26.7 m away from the ellipsoidal answer.

## Following a great circle

This is the "direct" problem of geodesy: given a start, an initial bearing θ and a distance d, find the end point. On a sphere of radius R, with δ = d / R as the angle travelled:

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)

The main answer comes from Karney's solution of the direct problem on the WGS84 ellipsoid; the sphere's answer is shown underneath with the gap between the two. Over 10 km they differ by a few tens of metres; over thousands of kilometres, by kilometres.

The traveller holds the starting bearing only for a moment, then follows the great circle, so the **final bearing** is usually different. Start on the equator heading 090° and you stay on it; head north and you cross the pole and come back down the other side, as the "12,000 km north" example shows. Uses include dead reckoning, laying out a search area or checking a waypoint someone described as "12 km on 210° from the summit".

## Bearings

- Enter a true bearing, from true north.
- Magnetic bearing + east declination = true bearing.
- Compass points: NE = 045°, SSW = 202.5°.

## Sources

- Karney CFF. Algorithms for geodesics. *Journal of Geodesy* 87 (2013), 43–55. [doi:10.1007/s00190-012-0578-z](https://doi.org/10.1007/s00190-012-0578-z), via [GeographicLib](https://geographiclib.sourceforge.io/).
- Veness C. [Destination point given distance and bearing from start point](https://www.movable-type.co.uk/scripts/latlong.html), Movable Type Scripts; its worked example (53°19′14″N 001°43′47″W, 096°01′18″, 124.8 km) is in this page's tests.

## Questions

**Can I enter a magnetic bearing?**

Convert it first: add the local declination if it is east, subtract it if it is west. The calculator itself works with true bearings.

**Is the destination on a straight line on my map?**

Over short distances, practically yes. Over long ones, a great circle looks curved on most flat maps; only a gnomonic projection draws it straight.

**What if the route crosses the 180° meridian?**

Nothing special: longitudes are wrapped back into −180° to +180°, so a route east from Fiji ends at a negative (western) longitude.