# Circular arc

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Part of a circle between two points

A [circular sector](/source/Circular_sector) is shaded in green.  Its curved boundary of length L is a circular arc.

A **circular arc** is the [arc](/source/Arc_(geometry)) of a [circle](/source/Circle) between a pair of distinct [points](/source/Point_(geometry)). If the two points are not directly opposite each other, one of these arcs, the **minor arc**, [subtends](/source/Subtended_angle) a [central angle](/source/Central_angle) that is less than [π](/source/Pi) [radians](/source/Radian) (180 [degrees](/source/Degree_(angle))); and the other arc, the **major arc**, subtends an [angle](/source/Angle) greater than π radians. The arc of a circle is defined as the part or segment of the [circumference](/source/Circumference) of a circle. A straight line that connects the two ends of the arc is known as a *[chord](/source/Chord_(geometry))* of a circle. If the [length](/source/Length) of an arc is exactly half of the circle, it is known as a *[semicircular arc](/source/Semicircle)*.

## Length

See also: [Arc length § Arcs of circles](/source/Arc_length#Arcs_of_circles)

The length (more precisely, [arc length](/source/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](/source/Central_angle)** — is

- L = θ r . {\displaystyle L=\theta r.}

This is because

- L c i r c u m f e r e n c e = θ 2 π . {\displaystyle {\frac {L}{\mathrm {circumference} }}={\frac {\theta }{2\pi }}.}

Substituting in the circumference

- L 2 π r = θ 2 π , {\displaystyle {\frac {L}{2\pi r}}={\frac {\theta }{2\pi }},}

and, with *α* being the same angle measured in degrees, since *θ* = ⁠*α*/180⁠π, the arc length equals

- L = α π r 180 . {\displaystyle L={\frac {\alpha \pi r}{180}}.}

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](/source/Angle) in degrees/360° = *L*/circumference.

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

- 60 360 = L 24 360 L = 1440 L = 4. {\displaystyle {\begin{aligned}{\frac {60}{360}}&={\frac {L}{24}}\\[6pt]360L&=1440\\[6pt]L&=4.\end{aligned}}}

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

- y = r 2 − x 2 . {\displaystyle y={\sqrt {r^{2}-x^{2}}}.}

Then the arc length from x = a {\displaystyle x=a} to x = b {\displaystyle x=b} is

- L = r [ arcsin ⁡ ( x r ) ] a b . {\displaystyle L=r{\Big [}\arcsin \left({\frac {x}{r}}\right){\Big ]}_{a}^{b}.}

## Sector area

Main article: [Circular sector § Area](/source/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

- A = r 2 θ 2 . {\displaystyle A={\frac {r^{2}\theta }{2}}.}

The area *A* has the same proportion to the [circle area](/source/Circle#Area_enclosed) as the angle *θ* to a full circle:

- A π r 2 = θ 2 π . {\displaystyle {\frac {A}{\pi r^{2}}}={\frac {\theta }{2\pi }}.}

We can cancel π on both sides:

- A r 2 = θ 2 . {\displaystyle {\frac {A}{r^{2}}}={\frac {\theta }{2}}.}

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

- A = 1 2 r 2 θ . {\displaystyle A={\frac {1}{2}}r^{2}\theta .}

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

- A = α 360 π r 2 . {\displaystyle A={\frac {\alpha }{360}}\pi r^{2}.}

## Segment area

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

- 1 2 r 2 ( θ − sin ⁡ θ ) . {\displaystyle {\frac {1}{2}}r^{2}(\theta -\sin \theta ).}

To get the area of the [arc segment](/source/Circle#Terminology), 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 A {\displaystyle A} . See [Circular segment](/source/Circular_segment) for details.

## Radius

[The product](/source/Product_(mathematics)) of the [line segments](/source/Line_segment) 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

        C
        D
        =

              A
              P
              ⋅
              P
              B

              C
              P

        +
        C
        P

    {\displaystyle CD={\frac {AP\cdot PB}{CP}}+CP}

Using the [intersecting chords theorem](/source/Power_of_a_point#Intersecting_secants_theorem,_intersecting_chords_theorem) (also known as [power of a point](/source/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](/source/Chord_(geometry)) 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 ⁠*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](/source/Sagitta_(geometry)) of the arc, *H*, and the other part is the remainder of the diameter, with length 2*r* − *H*. Applying the intersecting chords theorem to these two chords produces

- H ( 2 r − H ) = ( W 2 ) 2 , {\displaystyle H(2r-H)=\left({\frac {W}{2}}\right)^{2},}

whence

- 2 r − H = W 2 4 H , {\displaystyle 2r-H={\frac {W^{2}}{4H}},}

so

- r = W 2 8 H + H 2 . {\displaystyle r={\frac {W^{2}}{8H}}+{\frac {H}{2}}.}

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

## See also

- [Biarc](/source/Biarc)

- [Circle of a sphere](/source/Circle_of_a_sphere)

- [Circular-arc graph](/source/Circular-arc_graph)

- [Circular interpolation](/source/Circular_interpolation)

- [Lemon (geometry)](/source/Lemon_(geometry))

- [Meridian arc](/source/Meridian_arc)

- [Circumference](/source/Circumference)

- [Circular motion](/source/Circular_motion)

- [Tangential speed](/source/Tangential_speed)

## External links

Wikimedia Commons has media related to [Circle arcs](https://commons.wikimedia.org/wiki/Category:Circle_arcs).

- [Table of contents for Math Open Reference Circle pages](http://www.mathopenref.com/tocs/circlestoc.html)

- [Math Open Reference page on circular arcs](http://www.mathopenref.com/arc.html) With interactive animation

- [Math Open Reference page on Radius of a circular arc or segment](http://www.mathopenref.com/arcradius.html) With interactive animation

- [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Arc"](https://mathworld.wolfram.com/Arc.html). *[MathWorld](/source/MathWorld)*.

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Adapted from the Wikipedia article [Circular arc](https://en.wikipedia.org/wiki/Circular_arc) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Circular_arc?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.