# Arc

An arc is a segment of a curve, most often a circle. In the strictest definition, an arc is a segment of a curve in a plane. Examples include segments of geometrical forms such as circles, ellipses, and parabolas, as well as irregular arcs defined by analytical functions.

Arcs of circles can be classified by size. A minor arc is one whose length is shorter than one-half of the circumference of a circle. A major arc is one whose length is longer than one half of the circumference of a circle. An arc whose length is exactly one-half of the circumference of the circle is simply called a semi-circle. The line connecting the endpoints of a major arc or minor arc is called a chord.

Angles subtended by circles can be classified by the location of the vertex. One important type of angle has the vertex located at the circumference. An angle whose vertex is at the center of the circle is called a central angle. Each specific central angle is subtended by only one arc, but each arc subtends infinitely many angles.

An arc of a circle can be measured by length along the circumference, or in terms of the angle subtended by the arc. A theorem of geometry states that the measure of the central angle of the circle is the measure of corresponding arc. If the arc lies on a circle of radius r and subtends a central angle (LA) measured in degrees, then the length of the arc is given by b = 2πr(LA/360).

In the case of irregular arcs, the length can be determined using calculus and differential geometry.

Kristin Lewotsky

## KEY TERMS

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Circumference

—The line defined by the collection of points at a distance r from the center of a circle.

Subtend

—Intersect.

Vertex

—The point at which the two sides of an angle meet.