EMT 725
Philippa M. Rhodes
Derive a formula to find the area of the segment of a circle.
A segment of a circle is the union of arc ACB, the chord AB, and the points of the interior of the circle that lie on the same side of segment AB as point C.
One way to derive the formula is to use the area of the sector that contains the segment and subtract the area of the isoceles triangle.
Let r = radius and h = altitude of the isosceles triangle. Notice that the isoceles triangle is two congruent right triangles.
So, 
and the area of the isosceles triangle in terms of r and h is At = 
.
Let the angle opposite side b equal p . Thus, tan p = b / h.
The area of the sector is 
.
Therefore, the area of a segment of a circle is
