Chord

The Line Segment joining two points on a curve. The term is often used to describe a Line Segment whose ends lie on a Circle. In the above figure, $r$ is the Radius of the Circle, $a$ is called the Apothem, and $s$ the Sagitta.

The shaded region in the left figure is called a Sector, and the shaded region in the right figure is called a Segment.

All Angles inscribed in a Circle and subtended by the same chord are equal. The converse is also true: The Locus of all points from which a given segment subtends equal Angles is a Circle.

Let a Circle of Radius $R$ have a Chord at distance $r$. The Area enclosed by the Chord, shown as the shaded region in the above figure, is then

$$A=2\int_0^{\sqrt{R^2-r^2}} x(y)\,dy.$$ (1)

But

$$y^2+(r+x)^2=R^2,$$ (2)

so

$$x(y)=\sqrt{R^2-y^2}-r$$ (3)

and

$$A = 2\int_0^{\sqrt{R^2-r^2}} (\sqrt{R^2-y^2}\,-r)\,dy$$

$$= \left[{y\sqrt{R^2-y^2}+R^2\tan^{-1}\left({y\over\sqrt{R^2-y^2}}\right)-2ry}\right]_0^{\sqrt{R^2-r^2}}$$

$$= r\sqrt{R^2-r^2}+R^2\tan^{-1}\left[{\left({R\over r}\right)^2-1}\right]-2r\sqrt{R^2-r^2}$$

$$= R^2\tan^{-1}\left[{\left({R\over r}\right)^2-1}\right]-r\sqrt{R^2-r^2}.$$ (4)

Checking the limits, when $r=R$, $A=0$ and when $r\to 0$,

$$A={\textstyle{1\over 2}}\pi R^2,$$ (5)

See also Annulus, Apothem, Bertrand's Problem, Concentric Circles, Radius, Sagitta, Sector, Segment