Area of a Circle Segment

Date: 04/18/99 at 16:21:19
From: Blair
Subject: Area of a segment

What I need help on is figuring out the formula of steps for how to 
figure out the area of a segment. For example, take a circle with an 
angle of 120 degrees. Where the two radii meet with the circle, a line 
is drawn to connect them. The area of the segment is the area from 
that line connecting the two radii, to the outer edge of the circle. 
How would you figure this out?

Date: 04/19/99 at 12:36:20
From: Doctor Peterson
Subject: Re: Area of a segment

Hi, Blair.

To find this area in general, you need to use trigonometric 
functions. Here's a place to find the formula in our FAQ (scroll down 
to Segment of a Circle):

http://mathforum.org/dr.math/faq/formulas/faq.circle.html   

In your case, since you know the angle, and your angle is one of the 
easy ones to work with, you can do without this. Here's how you can 
come up with a formula.

You want the area of a segment of a circle, between a chord and its 
arc:

             ***
        *           *
      *               *
     *                 *
    *                   *
    *         +---------*----
    *    r /  t   \     *  |
     *  /            \ *   |h
      *---------------* -----
        *###########*
             ***

You know the angle t and the radius r. Given that t is 120 degrees, 
you can find h, the altitude of the triangle between the two radii. 
(I'll leave that part for you to do.)

You can think of this area as a sector (bounded by the arc and two 
radii) minus a triangle (bounded by the radii and the chord):

    sector:
             ***
        *           *
      *               *
     *                 *
    *                   *
    *         +         *
    *      /######\     *
     *  /############\ *
      *###############*
        *###########*
             ***

    triangle:
             ***
        *           *
      *               *
     *                 *
    *                   *
    *         +         *----           +
    *      /######\     *  |            |   \ r
     *  /############\ *   |h          h|      \
      *---------------*------           +---------+
        *           *                        b
             ***

The triangle has height h, and base 2*b, where b is the other side of 
the right triangle with hypotenuse r and side h). So the area is

    A = 2*b * h / 2
      = h * sqrt(r^2 - h^2)

The angle at the top of the sector is t = 120 degrees. Since the area 
of a sector is proportional to the angle,

          t
    A = ----- * pi * r^2
         360

        pi * r^2 * t
      = ------------
            360

The area of the segment is the difference of these, or

        pi * r^2 * t
    A = ------------ - h * sqrt(r^2 - h^2)
            360

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/