Why Do Flights to America Fly Near the North Pole?

You can find the shortest route yourself, without any maths, if you have a globe (or a football and a pen).

Here is a puzzle small enough to solve in your head. Two towns are both at 60∘° north, on exactly opposite sides of the Earth: one at longitude 0∘° and one at 180∘°. Which is shorter, to fly due west along the 60∘° line of latitude, or to fly straight over the North Pole?

Along the line of latitude you cover half of a circle. The 60∘° circle is smaller than the equator: its radius is the Earth’s radius R times cos 60∘°, which is half of R. So half of it is

$$\tfrac12 \times 2\pi \times \tfrac{R}{2} = \tfrac{\pi R}{2} \approx 10{,}008\ \text{km}.$$

Over the pole you go from 60∘° up to 90∘° and down to 60∘° on the other side: 30∘° + 30∘° = 60∘° of a circle the full size of the Earth, a sixth of it:

$$\tfrac{60}{360} \times 2\pi R = \tfrac{\pi R}{3} \approx 6{,}672\ \text{km}.$$

Over the pole is a third shorter. And it is not a special trick of 60∘°. The same comparison at other latitudes:

Two towns at latitude, opposite sides	0∘°	30∘°	45∘°	60∘°	75∘°
Along the line of latitude	20,015 km	17,334 km	14,153 km	10,008 km	5,180 km
Over the North Pole	20,015 km	13,343 km	10,008 km	6,672 km	3,336 km

Only on the equator are the two the same. Everywhere else the line of latitude, which the wall map draws straight, is the longer way.

The towns don’t have to be on opposite sides. At 60∘° north, a quarter of the way round (90∘° of longitude apart) the line of latitude is 5,004 km and the shortest route 4,605 km, bending up to 67.8∘° north on the way. The farther apart the towns, the more the shortest route bends towards the pole, until, exactly opposite, it goes over it.

Cut a ball in half with a knife through its center, and the cut edge is the biggest circle you can draw on that ball. On the Earth, such a circle is called a great circle: a circle on the sphere whose center is the center of the sphere. The equator is one. Every line of longitude together with its partner on the other side (0∘° with 180∘°) is one. The other lines of latitude are not: the 60∘° circle’s center is on the Earth’s axis, far above the center of the Earth, and its radius is only half the Earth’s.

Through any two places that aren’t exactly opposite there is exactly one great circle: the plane through the two places and the Earth’s center cuts it out. And here is the fact that answers Anisha’s question:

The shortest way between two places on a sphere is along the great circle through them, the shorter of its two arcs. A pilot or a ship that follows it is said to follow the great-circle route.

Why the biggest circle wins

Why should the biggest circle give the shortest way? Take the two towns at 60∘° north again, A and B. Many circles on the globe pass through both: the 60∘° line of latitude is one, the great circle over the pole is another, and in between there are tilted circles of every size. Each one lies in a flat plane through A and B. Lay each circle flat on the page, and you can compare them in the widget.

The way from A to B is an arc, and an arc is a fraction of its circle, the fraction set by its angle at the center, θ, as in Grade 10’s Central angle and its arc:

$$\text{arc} = \frac{\theta}{360^\circ} \times 2\pi r.$$

As the circle grows, two things happen at once. Its whole length $2\pi r$ grows, which would make the arc longer. But the center moves away from A and B, so the angle θ they make there shrinks, and it shrinks faster. The smallest circle has $\theta = 180^\circ$ (A and B at the ends of a diameter: half the circle, 10,008 km); a circle of radius 0.7 R has $\theta \approx 91.2^\circ$ and an arc of 7,096 km; the great circle has $\theta = 60^\circ$ and an arc of 6,672 km. Why does the angle win? Look at the widget: each bigger circle’s arc lies inside the smaller one’s, between it and the straight line AB, and of two outward-bulging curves with the same ends, the one inside is always the shorter. The flattest arc the sphere allows is on its biggest circle.

So the great circle is shorter than every other circle through A and B. A route doesn’t have to be part of a circle, of course: it could wiggle. That the great circle also beats every wiggly route is true, and proved in university courses on the geometry of curved surfaces; the stretched thread is the experiment that shows it.

Why the wall map makes it look curved

A plain wall map, with its lines of latitude evenly spaced, draws every line of latitude as a straight line of the same length as the equator. But on the Earth the 60∘° line is only half as long as the equator, and the 80∘° line less than a fifth. So the map stretches the north sideways, more and more towards the top, until the North Pole, a single point, becomes the whole top edge. A route that hurries across the stretched-out north looks long and bent on the map, when on the Earth it is the short cut.

No flat map can keep every shortest route straight and keep shapes and sizes right; the site’s Flat Maps shows eleven ways of trying. But one map does make every great circle straight. Put a lamp at the very center of a glass globe and let it shine the coastlines onto a flat sheet touching the globe. Each great circle lies in a plane through the lamp, and a plane meets the flat sheet in a straight line. This is the gnomonic map (“Shortest routes” on Flat Maps). It can’t show even half the Earth, and it stretches shapes badly near its edge, but navigators have long used it to plan a long route: draw a straight line on it, then copy the line onto an ordinary chart. The widget’s Globe view, seen from straight above the middle of a route, shows the same thing: there the great circle is a straight line through the middle of the picture, and the map’s “straight” line bends.

Working out the distance

How did we get 12,975 km? Put the Earth’s center at the origin and measure in Earth radii. A place at latitude $\varphi$ and longitude $\lambda$ is then the point

$$(\cos\varphi\cos\lambda,\ \cos\varphi\sin\lambda,\ \sin\varphi),$$

which is 1 from the center. The angle θ between two such points, seen from the center, comes from the scalar product of Grade 10’s Optional Mathematics, $\vec a \cdot \vec b = |\vec a|\,|\vec b| \cos\theta$, in three dimensions. Multiply the two points coordinate by coordinate and add, and with $\cos\lambda_1\cos\lambda_2 + \sin\lambda_1\sin\lambda_2 = \cos(\lambda_2 - \lambda_1)$:

$$\cos\theta = \sin\varphi_1 \sin\varphi_2 + \cos\varphi_1 \cos\varphi_2 \cos(\lambda_2 - \lambda_1).$$

The distance is then the arc, $\frac{\theta}{360^\circ} \times 2\pi R$, with R = 6{,}371 km, the Earth’s mean radius.

Here is a walk you can do on a globe with a finger. Start at the North Pole. Go straight down the 0∘° line of longitude to the equator. Turn left, a right angle, and walk east along the equator for a quarter of the way round, to 90∘° E. Turn left again, another right angle, and walk straight north up the 90∘° line, back to the pole. You arrive at the pole at right angles to the line you left it by.

You have walked a triangle: three straight sides (each a great circle, the straightest a line on a sphere can be) and three right angles, 270∘° in all. On flat paper the angles of a triangle always add to 180∘°. On a sphere they don’t.

Play with it and you find three things. Every triangle’s angles add to more than 180∘°. The bigger the triangle, the more they add to: Kathmandu, Cape Town and Sydney make 274∘°, more than three right angles. And the two numbers at the bottom never disagree: the extra over 180∘°, divided by 720∘°, is always the triangle’s share of the whole sphere. This is Girard’s theorem:

$$\text{angle sum} - 180^\circ = 720^\circ \times \frac{\text{area of the triangle}}{\text{area of the sphere}}.$$

Check it on the three right angles: 270∘° − 180∘° = 90∘°, and 90∘° ÷ 720∘° = 1/8. Is the triangle an eighth of the sphere? The equator cuts the sphere into two halves, and the 0∘° and 90∘° lines cut the northern half into four equal quarters: yes, one eighth.

Why it is true: peeling the orange

The proof needs only one idea. Two great circles through a point A cross again at A′, the point exactly opposite, and between them they cut out a slice like a segment of an orange, called a lune. If the angle between them at A is α, the lune is $\alpha/360^\circ$ of the sphere, just as a slice of a round cake cut at angle α is $\alpha/360^\circ$ of the cake.

Now take a triangle with angles α, β, $\gamma$, and extend each of its sides to the whole great circle. At each corner there is a lune holding the triangle (Figure 20.1), and a matching lune on the far side holding the opposite triangle (the one made by the three opposite points A′, B′, C′, the same size as ABC). The three great circles cut the sphere into eight triangles: ABC, its twin, and six others. Each lune is two of them, its own triangle and the one across that triangle’s far side (the lune at A is ABC and A′BC). So the six lunes cover each of the six others once, and ABC and its twin three times each. As shares of the sphere, with the triangle’s share T:

$$2\left(\frac{\alpha}{360^\circ} + \frac{\beta}{360^\circ} + \frac{\gamma}{360^\circ}\right) = 1 + 4T.$$

The 1 is the sphere, and the 4T is the two extra layers over each of the two triangles. Multiply by 180∘° and rearrange:

$$\alpha + \beta + \gamma = 180^\circ + 720^\circ \times T.$$

That is Girard’s theorem, and it shows why the sum is always more than 180∘°: a triangle always has some area.

Small triangles are nearly flat

Turn it round: a small triangle has a tiny share of the sphere, so its angles add to only a hair over 180∘°. A triangle as big as Nepal (about 147,000 km²) has angles adding to 180∘° plus 0.21∘°, about 12.5 minutes of arc. That is why flat geometry works for fields and buildings: on the scale of a town, the Earth is flat for every practical purpose.

But surveys that mapped countries with chains of triangles, measuring angles from hilltop to hilltop, had to allow for it: even an equilateral triangle with 21.3 km sides has an extra of about 1 second of arc (a sixtieth of a minute). Legendre published a simple way to do it in 1787 (surveyors may have used it before him): take a third of the triangle’s extra off each angle and work as if on flat paper.

• Flight routes. Airlines connecting Asian and Middle Eastern cities with North America fly over the Arctic, and Emirates and Qatar Airways fly nonstop from Dubai and Doha to San Francisco, Seattle and Los Angeles, coming within a few degrees of the pole (as of 2026). Real routes aren’t exactly great circles: airlines also ride the jet stream, a fast wind high up, and avoid closed airspace. Since 2022 many airlines have avoided Russian airspace, and Japan Airlines’ London–Tokyo flight, sent over Greenland, Canada and Alaska instead, takes about four hours longer.
• Navigation charts. Navigators of ships and aircraft draw the shortest route as a straight line on a gnomonic chart, then copy it onto a Mercator chart to steer by.
• Surveying. Before satellites, national surveys corrected the angles of their triangles for the spherical extra, by Legendre’s rule, from about 1800 until the middle of the twentieth century.

• How Did Eratosthenes Measure the Earth? (Grades 6–8) The Earth’s size from a shadow and a central angle, the radius every number on this page uses.
• What Is the Shortest Way to the River and Back? (Grades 6–8) Shortest routes on flat ground, found with a mirror.
• How Long Is Nepal’s Border? Why the length of a wiggly line depends on the ruler you measure it with.
• Why Are There Only Five Perfect Dice? Another surprise from shapes that close up on themselves.
• Our World: Flat Maps Eleven maps of the round Earth, the gnomonic “Shortest routes” map among them, and what each keeps and breaks.
• Grade 10 Math: Central angle and its arc The rule arc = (θ ÷ 360∘°) × 2πr behind every distance here.