Asgor asked in Science & MathematicsMathematics · 1 month ago

# How can I calculate the aerial distance between the two cities?

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• 4 weeks ago

Open  "Google Maps" online. It can tell you the distance between two cities.

• Anonymous
1 month ago

Distance calculator calculate the distance between any cities or places and measure the distance in miles and kilometers. Air distance (also called great circle or orthodrome) is also drawn on the distance map below.

Enter the city name, location name or the location coordinates in lat long format (lat,long) and click measure button to measure the distance between cities or any two places in the world. Find the distance from cities in miles and kilometers for flying air distance.

• 1 month ago

What do you know about these cities? Do you know their latitude and longitude?

• 1 month ago

That problem in spherical geometry has historical significance ever since people realized the shortest sailing distance between two distant harbors is actually a curve, not a straight line.

Observe the spherical triangle in the picture. Let B and C are two cities, and A is one of the poles. Let latitudes and longitudes of B and C are known.

Obviously, angle A is Long(C)-Long(B)

Angle c is 90°-Lat(B)

Angle b is 90°-Lat(C)

If we could know angle "a", with that angle and radius of sphere we could find arc distance BC.

Law of Cosines on a sphere (I'll just write formula without proof)

cos A = (cos a - cos b cos c) / (sin b sin c)

from here

cos a = sin b sin c cos A + cos b cos c

a = arccos (sin b sin c cos A + cos b cos c)

If angle a is in radians, we just need to multiply it by the radius of sphere R to find the distance (arc length) BC

BC = a R

Example

Let's find the distance from New York City to Paris (Lindbergh's flight)

coordinates

New York City - 40°44′31″N ..... 73°35′56″W

Paris - : ............ 48°58′10″N .....   2°26′29″E

Angle A = 73°35′56″ + 2°26′29″ = 76° 02' 25'' = 76.040°

Angle c = 90° - 40°44′31″ = 49° 15' 29'' = 49.258°

Angle b = 90° - 48°58′10″ = 41° 1' 50'' = 41.031°

a = arccos (sin 41.031° sin 49.258° cos 76.040° + cos 41.031° cos 49.258°)

a = 52.242° = 0.91180 rad

We'll take average radius of Earth R=6371km

BC = 0.91180 * 6371 = 5809 km

The actual distance will slightly differ because Earth is flattened at the poles, not a perfect sphere.

For short distances, this method can lead to rounding errors because cosines of small angles are very close to 1. For close cities, it is, therefore, more practical to assume that Earth is flat, meridians are the parallel lines and we find distance through the Pythagorean Theorem.

• Audrey
Lv 6
1 month ago