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[[Image:Circle arc.svg|thumb|300px|A [[circular sector]] is shaded in green.  Its curved boundary of length L is a circular arc.]]

In [[geometry]], an '''arc''' (symbol: '''⌒''') is a [[closed set|closed]] segment of a [[differentiable]] [[curve]] in the [[two-dimensional manifold|two-dimensional plane]]; for example, a '''circular arc''' is a segment of a [[circle]], or of its [[circumference]] ([[Boundary (topology)|boundary]]) if the circle is considered to be a [[Disk (mathematics)|disc]]. If the arc is part of a [[great circle]] (or [[great ellipse]]), it is called a '''great arc'''.

Except where stated otherwise, the arcs discussed below are arcs of circles.

== Arc length ==
{{Main|Arc length}}

===Length of an arc of a circle===
The [[arc length|length]], ''L'', of an arc of a circle with radius <math>r</math> and subtending an angle <math>\theta\,\!</math> (measured in [[radian]]s) with the circle center — i.e., the '''[[central angle]]''' — equals <math>\theta r\,\!</math>.  This is because

:<math>\frac{L}{\mathrm{circumference}}=\frac{\theta}{2\pi}.\,\!</math>

Substituting in the circumference

:<math>\frac{L}{2\pi r}=\frac{\theta}{2\pi},\,\!</math>
and since <math>\theta=\frac{\alpha}{180}\pi,\,\!</math>
the arc length equals

:<math>L=\frac{\alpha\pi r}{180}.\,\!</math> 
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

:<math>\frac{60}{360} = \frac{L}{24}</math>

:<math>360L=1440</math>

:<math>L = 4</math>.

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

===Length of an arc of a parabola===
If a point '''X''' is located on a [[parabola]] which has focal length <math>f,</math> and if <math>p</math> is the [[perpendicular distance]] from '''X''' to the axis of symmetry of the parabola, then the lengths of arcs of the parabola which terminate at '''X''' can be calculated from <math>f</math> and <math>p</math> as follows, assuming they are all expressed in the same units.

:<math>h=\frac{p}{2}</math>

:<math>q=\sqrt{f^2+h^2}</math>

:<math>s=\frac{hq}{f}+f\ln\left(\frac{h+q}{f}\right)</math>

This quantity, <math>s</math>, is the length of the arc between '''X''' and the vertex of the parabola.<ref>In this calculation, the [[square-root]], '''''q''''', must be positive. The quantity '''ln(''a'')''' is the [[natural logarithm]] of&nbsp;''a'', i.e. its logarithm to base&nbsp;"e".</ref>

The length of the arc between '''X''' and the symmetrically opposite point on the other side of the parabola is <math>2s.</math>

The perpendicular distance, <math>p</math>, can be given a positive or negative sign to indicate on which side of the axis of symmetry '''X''' is situated. Reversing the sign of <math>p</math> reverses the signs of <math>h</math> and <math>s</math> without changing their absolute values. If these quantities are signed, '''the length of the arc between ''any'' two points on the parabola is always shown by the difference between their values of <math>s.</math>''' The calculation can be simplified by using the properties of logarithms:

:<math>s_1 - s_2 = \frac{h_1 q_1 - h_2 q_2}{f} +f \ln \left(\frac{h_1 + q_1}{h_2 + q_2}\right)</math>

This can be useful, for example, in calculating the size of the material needed to make a [[parabolic reflector]] or [[parabolic trough]].

This calculation can be used for a parabola in any orientation. It is not restricted to the situation where the axis of symmetry is parallel to the y-axis.

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

:<math>A=\frac{1}{2} r^2 \theta.</math>

The area <math>A</math> has the same proportion to the [[Circle#Area_enclosed|circle area]] as the angle <math>\theta</math> to a full circle:

:<math>\frac{A}{\pi r^2}=\frac{\theta}{2\pi}.</math>

We can get rid of a <math>\pi</math> on both sides:

:<math>\frac{A}{r^2}=\frac{\theta}{2}.</math>

By multiplying both sides by <math>r^2</math>, we get the final result:

:<math>A=\frac{1}{2} r^2 \theta.</math>

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

:<math>A=\frac{\alpha}{360} \pi r^2.</math>

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

:<math>\frac{1}{2} r^2 (\theta - \sin{\theta}).</math>

To get the area of the [[Circle#Further_terminology|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 <math>A</math>. See [[Circular segment]] for details.

== Arc radius ==

Using the [[Power of a point#Theorems|intersecting chords theorem]] (also known as [[power of a point]] or secant tangent theorem) it is possible to calculate the radius <math>r</math> of a circle given the height <math>H</math> and the width <math>W</math> of an arc:

Consider the chord with the same end-points as the arc. Its perpendicular bisector is another chord, which is a diameter of the circle. The length of the first chord is <math>W,</math> and it is divided by the bisector into two equal halves, each with length <math>\frac{W}{2}.</math> The total length of the diameter is <math>2r,</math> and it is divided into two parts by the first chord. The length of one part is the height of the arc, <math>H,</math> and the other part is the remainder of the diameter, with length <math>(2r-H).</math> Applying the intersecting chords theorem to these two chords produces:

:<math>H(2r-H)=\left(\frac{W}{2}\right)^2</math>

whence:

:<math>2r-H=\frac{W^2}{4H}</math>

so:

:<math>r=\frac{W^2}{8H}+\frac{H}{2}.</math>

==See also==
*[[Arc length]]
*[[Biarc]]
*[[Circular-arc graph]]
*[[Meridian arc]]
*[[Circumference]]
*[[Perimeter]]

*[[Arc (disambiguation)#Mathematics|Other meanings of arc]]
Similar shapes:
*[[Catenary]]

== External links ==
*[http://www.mathopenref.com/arc.html Definition and properties of a circular arc] With interactive animation
*[http://www.mathopenref.com/arcradius.html Radius of an arc or segment] With interactive animation
*[http://www.mathopenref.com/tocs/circlestoc.html A collection of pages defining arcs and their properties, with animated applets] Arcs, arc central angle, arc peripheral angle, central angle theorem and others.
*{{MathWorld | urlname=Arc | title=Arc}}

== Notes and references ==
{{reflist}}

[[Category:Curves]]