Formula and Method for Great-Circle Distance
A great circle is the largest possible circle that can be drawn on a sphere — its plane always passes through the sphere's center. The arc of a great circle between any two points is the shortest possible path between them along the surface, which is why it is the route ships and aircraft actually follow for long-distance travel. This calculator finds that distance, plus the compass bearings and midpoint, from two latitude/longitude coordinates using the haversine formula.
How the calculation works
Convert both points' latitude (φ) and longitude (λ) to radians. Compute the differences Δφ = φ2 − φ1 and Δλ = λ2 − λ1, then apply the haversine formula: a = sin²(Δφ/2) + cos φ1 · cos φ2 · sin²(Δλ/2), followed by c = 2 · atan2(√a, √(1−a)). Multiplying c by Earth's mean radius R (about 6,371 km) gives the great-circle distance d = R · c. The initial bearing (compass heading leaving point 1) is θ = atan2(sin Δλ · cos φ2, cos φ1 · sin φ2 − sin φ1 · cos φ2 · cos Δλ), normalized to 0–360°. The final bearing is found by running the same formula from point 2 to point 1 and reversing the direction by 180°. The midpoint is the point exactly halfway along the great-circle arc, found with a separate vector-based formula.
Common mistakes
- Degrees vs. radians: latitude and longitude are entered in degrees, but trigonometric functions need radians. Forgetting to convert (multiply by π/180) gives wildly wrong results.
- Sign conventions: longitude is negative west of the Prime Meridian and positive east; latitude is negative south of the equator and positive north. Entering the wrong sign can place a point on the opposite side of the planet.
- Great circle vs. straight line on a map: a great-circle route often looks curved on a flat (Mercator) map even though it is the shortest real-world path — that curvature is a map projection artifact, not a longer route.
Real-world applications
- Airlines and shipping routes use great-circle paths to minimize fuel and travel time on long-haul journeys.
- Celestial navigation and orienteering use bearings computed the same way to plot a course between waypoints.
- GIS, mapping, and geofencing software use the haversine formula to compute distances between GPS coordinates.
- Radio and satellite communications use great-circle bearings to aim antennas at the shortest path to a target location.
Frequently Asked Questions
What is a great-circle distance?
A great-circle distance is the shortest path between two points on the surface of a sphere, measured along the arc of the "great circle" that passes through both points — a circle whose plane cuts through the sphere's center. On Earth, this is shorter than a straight line drawn on a flat map (a rhumb line) for any two points that are not on the same meridian or the equator.
What formula does this calculator use?
It uses the haversine formula: a = sin²(Δφ/2) + cos φ1 · cos φ2 · sin²(Δλ/2), c = 2 · atan2(√a, √(1−a)), and distance = R · c, where φ1, φ2 are the latitudes, Δφ and Δλ are the differences in latitude and longitude in radians, and R is Earth's mean radius (about 6,371 km).
What is the difference between a great circle and a rhumb line?
A great circle is the shortest path between two points on a sphere and generally appears curved on a flat map. A rhumb line (loxodrome) crosses every meridian at the same angle, so it looks straight on a Mercator map, but it is usually longer than the great-circle route, especially over long distances.
How accurate is the haversine formula?
The haversine formula treats Earth as a perfect sphere, so it is accurate to within about 0.5% of the true distance, which uses an ellipsoidal model (like Vincenty's formulae) for higher precision. For most navigation, aviation, and mapping purposes, the haversine result is more than accurate enough.
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 |