Free Azimuth Calculator

Azimuth Calculator: Free Bearing and Distance Between Two Points

This free online azimuth calculator – also called a bearing calculator – determines the bearing (azimuth) and the great‑circle distance between any two geographical points given their latitude and longitude coordinates. Whether you need to plan a route, align a satellite dish, or find the shortest path between cities, the tool applies the Haversine formula to deliver reliable results. Important: The calculator works exclusively with geographic coordinates in decimal degrees; it is not designed for general spherical coordinates.

What Is Azimuth?

Azimuth is the horizontal angle measured clockwise from True North. In cartographic terms, it is the angle formed between the line from your location to the North Pole and the line from your location to the target direction.

Common reference points:

• Directly east → $90^\circ$
• Directly south → $180^\circ$
• Directly west → $270^\circ$

In astronomy, azimuth pairs with altitude to describe the position of celestial objects: azimuth gives the compass direction, while altitude measures the vertical angle from the horizon ($0^\circ$) to the zenith ($90^\circ$). The point opposite the zenith is the nadir.

Coordinate Format

The tool expects all coordinates in decimal degrees. For example:

• New York City: $40.7128^\circ$ N, $74.0060^\circ$ W → $40.7128$, $-74.0060$
• Sydney: $-33.8688^\circ$ S, $151.2093^\circ$ E → $-33.8688$, $151.2093$

If you have degrees–minutes–seconds (DMS) values, convert them to decimal degrees before entering.

Distance Formula: Haversine

Because Earth is approximately spherical, the shortest path along its surface is a great‑circle arc. The Haversine formula calculates this great‑circle distance $d$:

$$d = 2R \cdot \arcsin\left( \sqrt{ \sin^{2}\!\left(\frac{\Delta\phi}{2}\right) + \cos\phi_1 \cdot \cos\phi_2 \cdot \sin^{2}\!\left(\frac{\Delta\lambda}{2}\right) } \right)$$

Where:

• $\phi_1, \phi_2$ = latitudes of the start and end points (positive for North, negative for South)
• $\lambda_1, \lambda_2$ = corresponding longitudes (positive for East, negative for West)
• $\Delta\phi = \phi_2 - \phi_1$, $\Delta\lambda = \lambda_2 - \lambda_1$
• $R = 6,\!371\ \text{km}$ (Earth’s mean radius)

All angular inputs must be in decimal degrees; the calculation converts them to radians internally.

Azimuth (Bearing) Formula

The initial heading (azimuth) from point 1 to point 2 is obtained using the two‑argument arctangent function:

$$\theta = \operatorname{atan2}\!\big( \sin\Delta\lambda \cdot \cos\phi_2,\ \cos\phi_1 \cdot \sin\phi_2 - \sin\phi_1 \cdot \cos\phi_2 \cdot \cos\Delta\lambda \big)$$

The result $\theta$ is in radians. Convert to degrees and add $360^\circ$ if needed to obtain a value between $0^\circ$ and $360^\circ$ (clockwise from North). This is the initial bearing – the direction you must head from the start to follow the great‑circle route.

Example: London to Rio de Janeiro

Let’s apply the formulas to a well‑known pair of cities.

• Start: London, UK — $51.50^\circ$ N, $0.00^\circ$
• Destination: Rio de Janeiro, Brazil — $22.97^\circ$ S, $43.18^\circ$ W

Using the Haversine and azimuth formulas (or this calculator) we obtain:

• Great‑circle distance: ≈ $9,\!280\ \text{km}$
• Initial azimuth: ≈ $219^\circ$ (i.e., about southwest)

From London you would set your compass to about $219^\circ$ and travel roughly 9,280 km along the great‑circle path to reach Rio.

Practical Applications

Azimuth and distance calculations are useful in:

• Navigation (air, sea, land) – determining courses between waypoints
• Satellite dish alignment – the azimuth tells you how far to rotate the antenna horizontally
• Astronomy – azimuth and altitude together pinpoint celestial objects
• GIS & surveying – computing bearings and distances for mapping and analysis

This free azimuth calculator online makes it easy to obtain the bearing and great‑circle distance between any two coordinates. Simply enter the latitude and longitude of both points in decimal degrees, and the tool instantly returns the azimuth (bearing) from the first point to the second along with the distance.