Great arc

In Euclidean geometry, an arc (symbol: ⌒) is a closed segment of a differentiable curve. A common example in the plane (a two-dimensional manifold), is a segment of a circle called a circular arc. In space, if the arc is part of a great circle (or great ellipse), it is called a great arc.

Every pair of distinct points on a circle determines two arcs. If the two points are not directly opposite each other, one of these arcs, the minor arc, will subtend an angle at the centre of the circle that is less than π radians (180 degrees), and the other arc, the major arc, will subtend an angle greater than π radians.

The length (more precisely, arc length), L, of an arc of a circle with radius r and subtending an angle θ (measured in radians) with the circle center — i.e., the central angle — equals θr. This is because

Substituting in the circumference

and, with α being the same angle measured in degrees, since θ = α/180π, the arc length equals

A practical way to determine the length of an arc in a circle is to plot two lines from the arc's endpoints to the center of the circle, measure the angle where the two lines meet the center, then solve for L by cross-multiplying the statement:

For example, if the measure of the angle is 60 degrees and the circumference is 24 inches, then

This is so because the circumference of a circle and the degrees of a circle, of which there are always 360, are directly proportional.

The area of the sector formed by an arc and the center of a circle (bounded by the arc and the two radii drawn to its endpoints) is

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