Understanding Azimuth and Great-Circle Bearing
Azimuth is the horizontal direction to a point, expressed as a compass angle measured clockwise from true north: 0° points north, 90° points east, 180° points south, and 270° points west. Given the latitude and longitude of two points, this calculator finds the initial azimuth — the compass heading you would face at the start of the great-circle (shortest-path) route between them — along with the return bearing and the distance itself.
The forward-azimuth formula
For a starting point A (latitude φ1, longitude λ1) and a destination B (latitude φ2, longitude λ2), the standard spherical-trigonometry bearing formula is:
θ = atan2( sin(Δλ) · cos(φ2), cos(φ1) · sin(φ2) − sin(φ1) · cos(φ2) · cos(Δλ) )
where Δλ = λ2 − λ1. The result θ is converted from radians to degrees and normalized to the 0–360° range to give the forward azimuth. The back azimuth (the bearing you would need at B to head back to A) is found by running the same formula with the two points swapped — it is generally not simply the forward azimuth plus 180°, because a great-circle path curves relative to a flat map except along the equator or a meridian.
Great-circle distance
Distance is computed with the haversine formula, which also assumes a spherical Earth of mean radius 6,371 km:
a = sin²(Δφ/2) + cos(φ1) · cos(φ2) · sin²(Δλ/2), then distance = 2R · atan2(√a, √(1−a))
This treats Earth as a perfect sphere, which is accurate to within roughly 0.5% of the true ellipsoidal distance — precise enough for navigation, antenna aiming, and general planning, though surveying work that needs millimeter accuracy should use an ellipsoidal (WGS84) model instead.
How to read the result
- Enter latitude as a signed decimal degree (north positive, south negative) and longitude the same way (east positive, west negative).
- The forward azimuth is the compass heading to leave Point A on; the compass direction card converts it to a familiar 16-point label like NE or WSW.
- The back azimuth tells you the heading to leave Point B on if retracing the same great-circle route toward Point A.
- If Point A and Point B are the same location, azimuth is undefined — the calculator will flag this rather than return a meaningless angle.
Frequently Asked Questions
What is azimuth?
How is azimuth calculated between two coordinates?
Why are the forward and back azimuth different by more than 180 degrees?
What Earth model does this calculator use?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |