Qibla Calculation

The Qibla is the direction of prayer toward the Kaaba in Mecca. PrayCalc calculates the precise great-circle bearing from any location on Earth using spherical trigonometry.

The great-circle bearing

The shortest path between two points on a sphere follows a great circle — the intersection of a sphere with a plane through its center. Qibla direction is the initial bearing of the great-circle path from the observer's location to the Kaaba.

Kaaba coordinates: 21.3891° N, 39.8579° E (Al-Masjid al-Haram, Mecca)

The bearing formula uses the spherical law of cosines adapted to forward azimuths:

θ = atan2(sin(Δλ) · cos(φ₂),
          cos(φ₁) · sin(φ₂) − sin(φ₁) · cos(φ₂) · cos(Δλ))

Where:

• φ₁ = observer latitude in radians
• φ₂ = Kaaba latitude in radians (21.3891° × π/180)
• Δλ = difference in longitude in radians (Kaaba longitude − observer longitude)
• θ = bearing in radians (convert to degrees, normalize to 0°–360°)

Step-by-step calculation

Example: Observer at New York City (40.7128° N, 74.0060° W)

1. Convert to radians:

   • φ₁ = 40.7128 × π/180 = 0.7105 rad
   • φ₂ = 21.3891 × π/180 = 0.3733 rad
   • Δλ = (39.8579 − (−74.0060)) × π/180 = 113.8639 × π/180 = 1.9872 rad

2. Calculate components:

   • sin(Δλ) = sin(1.9872) = 0.9152
   • cos(φ₂) = cos(0.3733) = 0.9311
   • cos(φ₁) = cos(0.7105) = 0.7603
   • sin(φ₂) = sin(0.3733) = 0.3647
   • sin(φ₁) = sin(0.7105) = 0.6497
   • cos(Δλ) = cos(1.9872) = −0.4027

3. Numerator: sin(Δλ) · cos(φ₂) = 0.9152 × 0.9311 = 0.8522

4. Denominator: cos(φ₁) · sin(φ₂) − sin(φ₁) · cos(φ₂) · cos(Δλ) = (0.7603 × 0.3647) − (0.6497 × 0.9311 × (−0.4027)) = 0.2773 − (−0.2439) = 0.5212

5. θ = atan2(0.8522, 0.5212) = 58.5°

Result: From New York City, the Qibla is approximately 58.5° (NE)

This matches the well-known fact that Qibla from North America points northeast over the Atlantic and Europe — not southeast as the flat-map "south toward Arabia" intuition suggests.

Distance to Mecca

PrayCalc also calculates the surface distance from the observer to the Kaaba using the Haversine formula, which is numerically stable for small distances (unlike the spherical law of cosines):

a = sin²(Δφ/2) + cos(φ₁) · cos(φ₂) · sin²(Δλ/2)
c = 2 · atan2(√a, √(1−a))
d = R · c

Where R = 6,371.0 km (Earth's mean radius) and Δφ = φ₂ − φ₁.

From New York City:

• Δφ = (21.3891 − 40.7128) × π/180 = −0.3372 rad
• Δλ = 1.9872 rad (as above)
• a = sin²(−0.1686) + cos(0.7105) · cos(0.3733) · sin²(0.9936)
• a = 0.02834 + 0.7603 · 0.9311 · 0.6978 ≈ 0.5225
• d = 6371 · 2 · atan2(0.7228, 0.7228) ≈ 9,141 km

Compass direction

PrayCalc converts the bearing angle to a cardinal direction string for display:

The Qibla compass in the app shows a live heading synchronized with the device's magnetometer via the Device Orientation API. The compass needle rotates to always point toward Mecca.

Accuracy and limitations

Earth's shape: The calculation assumes a perfect sphere. The Earth is an oblate spheroid (slightly flattened at the poles), making equatorial radius ~0.3% larger than polar. For Qibla purposes, the spherical approximation introduces an error of less than 0.2° at worst-case latitudes — well within the width of a typical prayer rug.

True vs. magnetic north: The bearing is relative to true geographic north, not magnetic north. Magnetic north drifts annually (currently about 50km/year) and varies by location by up to 20°+ in some regions. For high-precision applications, apply the local magnetic declination from the NOAA World Magnetic Model (WMM). For devotional use, the difference is not significant.

Coordinate precision: PrayCalc uses GPS-quality coordinates (7+ decimal places from browser geolocation) for maximum accuracy. The Kaaba coordinates are fixed at 21.38900° N, 39.85790° E.