Why the Qibla Follows a Great Circle, Not a Flat Map

From North America the Qibla points northeast because the direction to Makkah is measured along a great circle: the shortest path between two places on the surface of a globe. On a Mercator wall map, Makkah looks slightly south of east from New York. On the globe, the shortest route leaves New York heading about 58 degrees east of north, crosses the North Atlantic and curves down over Europe toward Arabia. The flat map shows a different kind of line, one that keeps a constant compass heading, and that line is longer.

Key Takeaways

• The usual Qibla bearing is the starting direction of the great circle from your location to the Kaaba.
• From New York it is about 58° from true north, from Chicago about 49°, and from Los Angeles about 24°, all northeast.
• A straight line on a Mercator map is a rhumb line (constant compass heading). It points slightly south of east from most of North America, but it is not the shortest path.
• A 1990s North American book argued for a southeast Qibla. The common Qibla algorithm, and the one used by the Qibla tool here, follows the great circle.

What the Qur'an asks, and what it leaves to measurement

The command to face the Sacred Mosque is in Surah al-Baqarah. In the Sahih International translation of the meaning on quran.com, Quran 2:144 says: "And wherever you [believers] are, turn your faces [i.e., yourselves] toward it [in prayer]."

The verse gives the destination, al-Masjid al-Haram. It does not give a compass number. Turning "toward" a place thousands of kilometres away is a question of geometry, and Muslim scholars treated it that way for centuries. R.H. van Gent of Utrecht University notes on his Qibla page that the problem occupied the greatest astronomers, geographers and mathematicians of the Islamic world, and that extensive tables were prepared for each latitude and longitude of the known world.

Great circles and rhumb lines

There are two common ways to draw a line from your city to Makkah. They give different directions.

A great circle is the line you get when a plane through the centre of the Earth slices the globe. The Naval Postgraduate School's notes on paths between points on Earth explain that on a spherical Earth a great circle is the shortest distance between two locations. Its compass bearing changes as you travel along it.

A rhumb line (also called a loxodrome) is a line of constant bearing. The same notes describe it as easy to follow but not the shortest distance on a long trip.

The two lines look very different depending on the map. Penn State's online course on map projections points out that on a Mercator map any rhumb line plots as a straight line with the correct compass angle. The Naval Postgraduate School notes give this as the reason the Mercator projection was invented. It is also why great-circle routes on it "do not look like the most direct ones," as the course's chapter on great circles and the gnomonic projection puts it. On a gnomonic map the reverse is true: every straight line is a great-circle arc.

So if you lay a ruler on a Mercator world map between Chicago and Makkah, you are drawing a rhumb line. The Qibla tools most people use draw the great circle instead.

Why the great circle bends north

Meridians, the lines of longitude, are not parallel on a globe. They meet at the poles. A path that cuts across them at a slant can save distance by crossing them at higher latitudes, where they are closer together. From mid-northern latitudes, the shortest route to a place far to the east therefore starts out heading north of the rhumb line and bows toward the pole before coming back down.

From North America the effect is large. Makkah is about 10,000 to 13,000 km from most of the United States and Canada, and far to the east, so the bow is large enough to swing the starting direction from slightly south of east to northeast.

Worked bearings from nine cities

The table compares the two directions from several cities. Bearings are measured clockwise from true north (0° is north, 90° east, 180° south).

Bearings use the standard spherical great-circle formula, initial bearing θ = atan2(sin Δλ · cos φ₂, cos φ₁ · sin φ₂ − sin φ₁ · cos φ₂ · cos Δλ), with the Kaaba at 21.4225241° N, 39.8261818° E and a mean Earth radius of 6,371 km. Each great-circle bearing matches the Qibla() function of the open-source adhan library to two decimal places. City coordinates are city-centre values, so a mosque across town can differ by a fraction of a degree. Repeating the calculation on the WGS 84 ellipsoid with Vincenty's inverse formula moves every bearing in the table by less than 0.2°.

A few things stand out:

• North America is where the two methods disagree most. From New York the gap is about 43°, from Los Angeles about 71°. The rhumb line from Los Angeles is also about 2,150 km longer.
• Anchorage faces almost due north. Alaska sits so far north and west that the shortest path to Makkah runs over the Arctic. The rhumb line points west-southwest, in nearly the opposite half of the compass.
• Distance alone does not set the gap. From Lagos and Jakarta the difference is under 4°, yet from London, only about 550 km farther than Lagos, it is about 15°. The gap also depends on how far the route runs east or west and how far north or south of Makkah the city lies.
• The southern hemisphere has its own version. From Sydney the great circle points just north of west, while a flat map suggests west-northwest.

A check from 1953

The great-circle direction is not a recent idea in North America. A 1953 newspaper account, reprinted in S. Kamal Abdali's 1997 paper The Correct Qibla, reports that the Islamic Center of Washington on Massachusetts Avenue was laid out facing 56°33′15″ east of true north. Visitors worried because a map line ran slightly south of east, and the article explains that they were thinking of a Mercator map. With the adhan library and coordinates for central Washington, D.C. (38.9072° N, 77.0369° W), the great-circle Qibla comes out at 56.6°, within about 0.01° of the 1953 figure.

The southeast position in North America

Not every community has accepted the great-circle answer. Van Gent's page notes an alternative method, followed by some Muslims, based on the rhumb line, a line of constant compass direction to the Kaaba.

In North America this became a public debate in the 1990s. Riad Nachef and Samir Kadi published a book in Philadelphia in 1414 AH (1993), The Substantiation of the People of Truth that the Direction of al-Qibla in the United States and Canada is to the Southeast. According to Abdali's summary of it, the book quoted classical jurists and appended statements from scientists, mainly geographers, who suggested identifying the Qibla with the rhumb line. Its authors stated that the region's mosques had faced southeast until a book by Abdali in the 1970s. Abdali disputed that history, pointing to the 1953 Washington mosque, and argued on geometric grounds for the great circle.

These are two parties' own accounts of the dispute. The scientific point, that the great circle is the shortest path and the rhumb line is not, is standard geodesy. Which line best fulfils the command to face the Kaaba is a religious question. If your mosque follows a different direction, ask its imam or a qualified scholar about the reasoning rather than changing it on your own.

Common questions

Does a flat-map direction make a big difference?

In North America, yes. The table shows gaps of about 43° to 100° between the two lines for the North American cities listed. From the other cities we checked (London, Paris, Lagos, Cape Town, Karachi, Delhi, Dhaka and Jakarta), the gap ranges from about 2° to 15°.

Is there anywhere the great circle has no single answer?

Yes, at the point exactly opposite Makkah on the globe, about 21.4° S and 140.2° W in the South Pacific. From there, every direction is a shortest path to the Kaaba. Near that point, small changes in position swing the Qibla bearing sharply.

Why does my compass reading differ from the bearing?

Qibla bearings are given from true north, and a magnetic compass points to magnetic north. The difference, called declination, depends on where you are. It is a separate correction from the choice of great circle or rhumb line, and it is covered on our sister site Everyday Muslim in True North and Magnetic North in a Qibla App.

Find the Qibla for your city

Use the Qibla direction tool to get the great-circle bearing from your exact location, calculated with the same great-circle formula as the adhan Qibla() function used in the table above. For a city in the US, start from the United States prayer times page, which lists states and cities with their prayer times. If you want to see how the sun's position sets the times themselves, read how prayer times are calculated.

When you set up a prayer space, mark the bearing against true north, then check it against a second reference such as the sun or a local mosque. If the result looks wrong on a flat map, that is usually the map, not the bearing.