arc (geometry)

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A circular sector is shaded in green. Its curved

boundary of length L is a circular arc.

Arc (geometry)From Wikipedia, the free encyclopedia

In geometry, an arc is a closed segment of adifferentiable curve in the two-dimensional plane;for example, a circular arc is a segment of the

circumference of a circle. If the arc occupies a greatcircle (or great ellipse), it is called a great-arc.

Contents

1 Arc Length2 Arc Area3 Arc Segment Area4 See also

5 External links

Arc Length

The length of an arc of a circle with radius r and subtending an angle (measured in radians) with thecircle 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 degrees has a size in radians given by

and so the arc length equals

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A practical way to determine the length of an arc in a circle is to plot two lines from the arc's endpointsto the center of the circle, measure the angle where the two lines meet the center, then solve for L bycross-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/24360L=1440L = 4".

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

Arc Area

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

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

We can get rid of a on both sides:

By multiplying both sides by r 2, we get the final result:

Using the conversion described above, we find that the area of the sector for a central angle measured indegrees is:

Arc Segment Area

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


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

See also

Arc length

Other meanings of arc Circular-arc graphMeridian arc

External links

Definition and properties of a circular arc (http://www.mathopenref.com/arc.html) Withinteractive animation

A collection of pages defining arcs and their properties, with animated applets(http://www.mathopenref.com/tocs/circlestoc.html) Arcs, arc central angle, arc peripheral angle,central angle theorem and others.

Weisstein, Eric W., "Arc (http://mathworld.wolfram.com/Arc.html) " from MathWorld.

Retrieved from "http://en.wikipedia.org/wiki/Arc_(geometry)"Categories: Geometry stubs | Curves

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