### Circular arc

In geometry, an **arc** (symbol: **⌒**) is a closed segment of a differentiable curve in the two-dimensional plane; for example, a **circular arc** is a segment of a circle, or of its circumference (boundary) if the circle is considered to be a 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.

## Contents

## Arc length

### Length of an arc of a circle

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

- $\backslash frac\{L\}\{\backslash mathrm\{circumference\}\}=\backslash frac\{\backslash theta\}\{2\backslash pi\}.\backslash ,\backslash !$

Substituting in the circumference

- $\backslash frac\{L\}\{2\backslash pi\; r\}=\backslash frac\{\backslash theta\}\{2\backslash pi\},\backslash ,\backslash !$

and solving for arc length, $L$, in terms of $\backslash theta\backslash ,\backslash !$ yields

- $L=\backslash theta\; r.\backslash ,\backslash !$

An angle of $\backslash alpha$ degrees has a size in radians given by

- $\backslash theta=\backslash frac\{\backslash alpha\}\{180\}\backslash pi,\backslash ,\backslash !$

and so the arc length equals

- $L=\backslash frac\{\backslash alpha\backslash pi\; r\}\{180\}.\backslash ,\backslash !$

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

- $\backslash frac\{60\}\{360\}\; =\; \backslash frac\{L\}\{24\}$

- $360L=1440$

- $L\; =\; 4$.

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

### Length of an arc of a parabola

If a point **X** is located on a parabola which has focal length $f,$ and if $p$ 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 $f$ and $p$ as follows, assuming they are all expressed in the same units.

- $h=\backslash frac\{p\}\{2\}$

- $q=\backslash sqrt\{f^2+h^2\}$

- $s=\backslash frac\{hq\}\{f\}+f\backslash ln\backslash left(\backslash frac\{h+q\}\{f\}\backslash right)$

This quantity, $s$, is the length of the arc between **X** and the vertex of the parabola.^{[1]}

The length of the arc between **X** and the symmetrically opposite point on the other side of the parabola is $2s.$

The perpendicular distance, $p$, 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 $p$ reverses the signs of $h$ and $s$ 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 $s.$** The calculation can be simplified by using the properties of logarithms:

- $s\_1\; -\; s\_2\; =\; \backslash frac\{h\_1\; q\_1\; -\; h\_2\; q\_2\}\{f\}\; +f\; \backslash ln\; \backslash left(\backslash frac\{h\_1\; +\; q\_1\}\{h\_2\; +\; q\_2\}\backslash right)$

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:

- $A=\backslash frac\{1\}\{2\}\; r^2\; \backslash theta.$

The area $A$ has the same proportion to the circle area as the angle $\backslash theta$ to a full circle:

- $\backslash frac\{A\}\{\backslash pi\; r^2\}=\backslash frac\{\backslash theta\}\{2\backslash pi\}.$

We can get rid of a $\backslash pi$ on both sides:

- $\backslash frac\{A\}\{r^2\}=\backslash frac\{\backslash theta\}\{2\}.$

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

- $A=\backslash frac\{1\}\{2\}\; r^2\; \backslash theta.$

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

- $A=\backslash frac\{\backslash alpha\}\{360\}\; \backslash pi\; r^2.$

## Arc segment area

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

- $\backslash frac\{1\}\{2\}\; r^2\; (\backslash theta\; -\; \backslash sin\{\backslash theta\}).$

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.

## Arc radius

Using the 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 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 $W,$ and it is divided by the bisector into two equal halves, each with length $\backslash frac\{W\}\{2\}.$ The total length of the diameter is $2r,$ and it is divided into two parts by the first chord. The length of one part is the height of the arc, $H,$ and the other part is the remainder of the diameter, with length $(2r-H).$ Applying the intersecting chords theorem to these two chords produces:

- $H(2r-H)=\backslash left(\backslash frac\{W\}\{2\}\backslash right)^2$

whence:

- $2r-H=\backslash frac\{W^2\}\{4H\}$

so:

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

## See also

Similar shapes:

## External links

- Definition and properties of a circular arc With interactive animation
- Radius of an arc or segment 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.
- MathWorld.