{{Short description|Part of a circle between two points}} [[Image:Circle arc.svg|thumb|upright=1.25|A [[circular sector]] is shaded in green. Its curved boundary of length L is a circular arc.]]

A '''circular arc''' is the [[arc (geometry)|arc]] of a [[circle]] between a pair of distinct [[Point (geometry)|points]]. If the two points are not directly opposite each other, one of these arcs, the '''minor arc''', [[subtended angle|subtends]] a [[central angle]] that is less than [[Pi|{{pi}}]] [[radian]]s (180 [[Degree (angle)|degrees]]); and the other arc, the '''major arc''', subtends an [[angle]] greater than {{pi}} radians. The arc of a circle is defined as the part or segment of the [[circumference]] of a circle. A straight line that connects the two ends of the arc is known as a ''[[chord (geometry)|chord]]'' of a circle. If the [[length]] of an arc is exactly half of the circle, it is known as a ''[[semicircle|semicircular arc]]''.

==Length== {{See also|Arc length#Arcs of circles}}

The length (more precisely, [[arc length]]) 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]]''' — is

:<math> L = \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, with ''α'' being the same angle measured in degrees, since ''θ''&nbsp;=&nbsp;{{sfrac|''α''|180}}{{pi}}, 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]] in degrees/360° = ''L''/circumference.

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

:<math>\begin{align} \frac{60}{360} &= \frac{L}{24} \\[6pt] 360L &= 1440 \\[6pt] L &= 4. \end{align}</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.

The upper half of a circle can be parameterized as

:<math>y=\sqrt{r^2-x^2}.</math>

Then the arc length from <math>x=a</math> to <math>x=b</math> is

:<math>L=r\Big[\arcsin \left(\frac{x}{r}\right)\Big]^b_a.</math>

== Sector area == {{main|Circular sector#Area}}

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

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

The area ''A'' has the same proportion to the [[Circle#Area enclosed|circle area]] as the angle ''θ'' to a full circle:

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

We can cancel {{pi}} on both sides:

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

By multiplying both sides by ''r''{{isup|2}}, 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>

== Segment area ==

The area of the shape bounded by the arc and the straight line between its two end points is

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

To get the area of the [[Circle#Terminology|arc segment]], we need to subtract the area of the triangle, determined by the circle's center and the two end points of the arc, from the area <math>A</math>. See [[Circular segment]] for details.

== Radius ==

[[File:Circle with some chords.jpg|thumb|upright=1.25|[[Product (mathematics)|The product]] of the [[line segment]]s AP and PB equals the product of the line segments CP and PD. If the arc has a width AB and height CP, then the circle's diameter <math>CD=\frac{AP \cdot PB}{CP}+CP</math>]]

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

Consider the [[Chord (geometry)|chord]] with the same endpoints as the arc. Its perpendicular bisector is another chord, which is a diameter of the circle. The length of the first chord is ''W'', and it is divided by the bisector into two equal halves, each with length {{sfrac|''W''|2}}. The total length of the diameter is 2''r'', and it is divided into two parts by the first chord. The length of one part is the [[Sagitta (geometry)|sagitta]] of the arc, ''H'', and the other part is the remainder of the diameter, with length 2''r''&nbsp;−&nbsp;''H''. 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>

The arc, chord, and sagitta derive their names respectively from the Latin words for [[Bow (weapon)|bow, bowstring, and arrow]].

==See also== *[[Biarc]] *[[Circle of a sphere]] *[[Circular-arc graph]] *[[Circular interpolation]] *[[Lemon (geometry)]] *[[Meridian arc]] *[[Circumference]] *[[Circular motion]] *[[Tangential speed]]

== External links == {{Commons category|Circle arcs}} *[http://www.mathopenref.com/tocs/circlestoc.html Table of contents for Math Open Reference Circle pages] *[http://www.mathopenref.com/arc.html Math Open Reference page on circular arcs] With interactive animation *[http://www.mathopenref.com/arcradius.html Math Open Reference page on Radius of a circular arc or segment] With interactive animation *{{MathWorld | urlname=Arc | title=Arc}}

[[Category:Circles]] [[Category:Curves]]