Finding the Area of an Arc

Date: 1/23/96 at 13:55:33
From: Sweet Home Middle School
Subject: Area of arc

Dear Dr. Math,

When you draw a circle and make a chord from one point to 
another, how would you find the area of that arc (formula)?

TIA for your answer!             
Sincerely,

Alyssa Ruffino, and Jillian Steinbroner
Sweet Home Middle School, Amherst, N.Y.  

Date: 6/5/96 at 18:35:12
From: Doctor Charles
Subject: Re: Area of arc

As far as I know the shape formed is called a segment.  There are 
two segments formed by this operation: the major segment and the 
minor segment.  (The major one contains the centre of the circle.)

To find the area of a minor segment think of it as a sector (slice of 
cake) minus a triangle with vertices at the centre of the circle and at 
the end points of the chord.

Suppose the angle of the triangle at the centre of the circle is x 
degrees.  Then the area of the sector is:

          Area of sector = pi * r^2 * x/360

(This formula comes from the formula of the area of a circle 
multiplied by the fraction of the circle that you are actually looking 
at.)

The length of the chord is 2 * r * sin(x/2) but I shall call it 2L for
now.  (This comes from the definition of sin on the triangle formed by
splitting the triangle which I have described above into two right 
angled triangles by drawing the perpendicular bisector of the chord.)

Think of the chord as the base of the triangle. Then the height of the 
triangle is r * cos(x/2).

Area of triangle = 1/2 * base * height = 1/2 * 2L * r * cos(x/2)

                                       = L * r * cos(x/2)

So the area of the segment is the area of the sector minus the area of 
the triangle.

   Area of Segment = pi * r^2 * x/360 - L * r * cos(x/2)
                   = pi * r^2 * x/360 - r^2 * sin(x/2) * cos (x/2)
                   = r^2 * (pi * x/360 - (sin x)/2)

Does this answer your query?

-Doctor Charles,  The Math Forum