Arc (geometry)

From Wikipedia, the free encyclopedia

In geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of the circumference of a circle. If the arc segment occupies a great circle (or great ellipse), it is considered a great-arc segment.

Contents 1 Arc Length 2 Arc Area 3 Arc Segment Area 4 See also 5 External links

[edit] Arc Length

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

Substituting in the circumference

and solving for arc length, L, in terms of yields

An angle of α degrees has a size in radians given by

and so 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:

measure of angle/360 = L/Circumference

For example, if the measure of the angle is 60 degrees and the Circumference is 24", then

60/360 = L/24

360L=1440

L = 4"

This is so due to the fact that the circumference of a circle and the degrees of a circle, of which there are always 360, are directly proportionate.

[edit] Arc Area

The area between an arc and the center of a circle is:

The area A has the same proportion to the circle area as the angle θ to a full circle:

We can get rid of a π on both sides:

By multiplying with r², we get the final result:

[edit] Arc Segment Area

The area of the shape limited by the arc and a straight line between the two end points is:

To get the area of the arc segment, we need to subtract the area of the triangle made up by the circle's center and the two end points of the arc from the area A. See Circular segment for details.

[edit] See also

• Arc length
• Other meanings of arc
• Circular-arc graph

[edit] External links

• Definition and properties of a circular arc With interactive animation
• A collection of pages defining arcs and their properties, with animated applets Arcs, arc central angle, arc peripheral angle, central angle theorem and others.
• Weistein, Eric W., "Arc" from MathWorld.