A circle basically is governed just by its radius. But in a segment of a circle a number of parameters are involved.

The length of the chord of the segment, its distance from the center of the circle, the length of arc of the segment of the circle which depends on its central angle, the height of the segment of the circle. The area of the circle depends on all these parameters. Let us derive the formulas for the measures of all these parameters in terms of the radius and the angle subtended by the arc at the center.

Refer to the following diagram.

The shaded area is the segment of the circle shown above. It is surrounded by the arc APB and below the chord AB of the circle.

First let us derive the formula for the length of the arc of the segment of the circle. Let ‘r’ be the radius of the circle. The arc APB is a part of the circumference of the circle and directly proportional to the angle (in radians) subtended by it at the center. Let that be θ (in radians) be that angle. That is, if ‘l’ is the length of the arc,

($\frac{l}{\theta}$) = ($\frac{C}{2 \pi}$) = ($\frac{2πr}{2π}$) = r, since C = circumference of the circle = 2πr

or, l = r $\theta$

Next let us try to establish a formula for ‘c’, the length of the chord.

From the center of the circle draw a perpendicular OR on AB and extend that to P, at the circumference.

As per the property of circles, AR = RB = $\frac{c}{2}$ and angle AOP = angle BOP = $\frac{\theta}{2}$

Consider the triangle AOR or BOR.

(

$\frac{c}{2r}$) = sin (

$\frac{\theta}{2}$) or,

c = 2r $\times$ sin ($\frac{\theta}{2}$) We can derive another formula for ‘c’. Applying cosine law for the triangle OAB,

AB

^{2} = OA

^{2} + OB

^{2} – 2 (OA)(OB)cos θ

or, c

^{2} = r

^{2} + r

^{2} – 2 (r)(r)cos θ = r

^{2} + r

^{2} – 2 r

^{2}cos θ = 2 r

^{2} – 2 r

^{2}cos θ = r

^{2}(2 - 2 cos θ)

or, c = r√(2 – 2 cos θ)

Thus, c = 2r * sin ($\frac{\theta}{2}$) = r√(2 – 2 cos θ) Let us now derive a formula for ‘h’, the height of the segment of a circle.

Consider again the triangle AOR or BOR.

OR (r) = cos (

$\frac{\theta}{2}$), or d = r $\times$ cos (

$\frac{\theta}{2}$)

Now,

h = RP = OP – OR = r - r $\times$ cos ($\frac{\theta}{2}$) = r[1 - cos ($\frac{\theta}{2}$)]The height ‘h’ can also be expressed in terms of ‘r’ and ‘c’.

In the triangle AOR or BOR, d

^{2} = r

^{2} – (

$\frac{c^2}{4}$)

or, d = √[ r

^{2} – (

$\frac{c^2}{4}$)]

so, h = r – d = r - √[ r

^{2} – (

$\frac{c^2}{4}$)]

Thus,

h = r[1 - cos ($\frac{\theta}{2}$)] = r - √[ r^{2} – ($\frac{c^2}{4}$)]Next, let us determine the angle θ.

(d) = r $\times$ cos (

$\frac{\theta}{2}$),

or, (

$\frac{\theta}{2}$) = cos

^{-1}(

$\frac{d}{2}$)

or,

θ = 2cos^{-1}($\frac{d}{r}$)Sometimes, a segment of a circle alone may be available or described. In such a case we may need to find the radius of the circular arc. We can do that by a formula that is derived below.

The obvious formula is r = d + h, but we can also derive a formula in terms of ‘c’ and ‘h’. It will be more useful because ‘d’ may not be known in most cases when ‘r’ is not known.

Again in the triangle AOR or BOR,

r

^{2} – d

^{2} = (

$\frac{c^2}{4}$)

or, r

^{2} – (r – h)

^{2} = (

$\frac{c^2}{4}$)

r

^{2} – (r

^{2} - 2r h + h

^{2}) = (

$\frac{c^2}{4}$)

2rh - h

^{2} = (

$\frac{c^2}{4}$)

2r - h = (

$\frac{c^2}{4h}$)

2r = h + (

$\frac{c^2}{4h}$) = [

$\frac{4h^2 + c^2}{4h}$]

or,

r = [$\frac{4h^2 + c^2}{8h}$]