Circular arc

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

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

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

${\displaystyle \frac{A}{\pi r^2}=\frac{\theta }{2\pi }.}$

We can cancel π on both sides:

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

By multiplying both sides by r², we get the final result:

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

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

${\displaystyle A=\frac{\alpha }{360}\pi r^2.}$

Segment area

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

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

To get the area of the 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 ${\displaystyle A}$. See Circular segment for details.

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 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 2r, and it is divided into two parts by the first chord. The length of one part is the sagitta 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

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

whence

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

so

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

The arc, chord, and sagitta derive their names respectively from the Latin words for bow, bowstring, and arrow.