Formula for the Length of a Chord

Date: Wed, 3 Jul 1996 10:11:02 -0400 (EDT) 
From: Anonymous
Subject: Length of chord

I'm trying to find a formula that will give me either 

the length of a chord when I know the circle radius and the distance 
from center to chordline 

OR 

the area of the resulting circle segment.

Date: Wed, 3 Jul 1996 12:07:22 -0400 (EDT) 
From: Dr. Anthony
Subject: Re: Length of chord

The first of these is easy, since you can use Pythagoras to find the 
length of half the chord (1/2*c) from a right-angled triangle. The 
hypotenuse is equal to the radius (= r) of the circle, and the 
perpendicular distance (= d) from the centre of the circle to the 
chord is also given. So we have: 

{(1/2)*c}^2 = r^2 - d^2

(1/4)*c^2 = r^2 - d^2

c = 2*sqrt(r^2-d^2)

The second is a little more complicated. We find the area of the 
segment by subtracting the area of triangle OAB from the area of 
the sector OAB of the circle, where O is the centre of the circle, 
and AB the chord. The angle AOB can be calculated from 
cos(1/2*AOB) = d/r, so 1/2*AOB = arccos(d/r) 

Therefore AOB = 2*arccos(d/r) (Give AOB in radians) 

Area of sector = (1/2)(AOB)r^2 (AOB in radians) 

Area of triangle AOB = (1/2)sin(AOB)*r^2 

Area of segment = r^2*[arccos(d/r) - (1/2)sin(2*arccos(d/r))] 

-Doctor Anthony, The Math Forum