Formula to find segment areas

• January 14th 2010, 07:03 AM
KJJ1974
Formula to find segment areas
Can anyone supply a formula to assist with the following problem please:

I need to work out the areas of segments of different circles (with known radii and with known perpendicular distances from the midpoint of the chord to the segment-part of the circle). I don't know the length of the chord or the angle of the sector that would be formed if lines were taken from the centre of the circle to the corners of the segment.

Hopefully this makes sense! Many thanks in advance. K
• January 14th 2010, 07:29 AM
Gusbob
Area of a segment is given by:

$A = \frac{1}{2}R^2(\theta - \sin \theta)$

Where R = Radius and $\theta$ = angle of the sector if lines were taken from the centre of the circle to the corners of the segment. Note that $\theta$ is in radians.

You do not have the angle yet, but you can find it from your known values (length of radius (R) and perpendicular distance from centre to midpoint of the chord (d)).

Examine the image below:

Now imagine a line d connecting the centre to the midpoint of the chord. This cuts the triangle into two smaller triangles, and also bisects angle $\theta$.
Look at one of the smaller triangles carefully. You'll see that you have a right angled triangle with two known sides R and d, which is the hypotenuse and the side adjacent to angle $\frac{1}{2} \theta$ respectively
Therefore $\cos \frac{1}{2} \theta = \frac{d}{R} \Rightarrow \theta = 2 \arccos \frac{d}{R}$