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# Arc Length

## Portion of a circle's circumference.

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How Far Is It to London?

Credit: Laura Guerin
Source: CK-12 Foundation

When an airplane flies from San Francisco to London, how many miles does it fly?

Why It Is Important

The distance covered across the globe is known to navigators as a great circle. It is the circle formed when you intersect the globe with a plane that passes through the center of the Earth and the two points on the circumference. If you can find the great circle, you can find the length of the arc between two locations.

We define positions on the globe in terms of latitude and longitude. London Heathrow Airport is at 51.5 degrees North and 0.5 degrees West. San Francisco International Airport is at 37.6 degrees North and 122.4 degrees West. If you draw a radius from each to the center of the Earth and calculate the angle between them, you will find that it is 77.5 degrees. The 77.5-degree arc between the two cities' airports covers approximately 5,350 miles.

Credit: Laura Guerin
Source: CK-12 Foundation

The image above depicts a more familiar map, but why do the maps look different? The problem comes from trying to represent a 3D sphere on a 2D piece of paper! This is the familiar Mercator map, which distorts areas near the poles. You can see that on the earlier map, the flight plan is a straight line, whereas on the Mercator map, it is curved. If you check the points on the ground over which the planes pass, however, you will see that they pass over the same places.

The study and science of map-making is called cartographyLearn more by watching this video: http://www.natgeoeducationvideo.com/film/1065/cartography

#### Explore More

The Earth is almost a sphere, with a radius of 3,960 miles.

1. What is the circumference of the Earth?
2. How can you use the circumference of the Earth to find that the distance between San Francisco and London is approximately 5,350 miles?

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1. [1]^ Credit: Laura Guerin; Source: CK-12 Foundation; License: CC BY-NC 3.0
2. [2]^ Credit: Laura Guerin; Source: CK-12 Foundation; License: CC BY-NC 3.0

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