Problem: Find the area of the segment if given r = radius and h= distance from the center to the chord.
Given r and h, we can construct an isosceles triangle as in the picture above. We also notice that the isosceles triangle is divided into two right triangles. We need to find the base of one of the right triangles.
Using the pythagorean theorem, we find that 
It follows that the area of the right triangle is 
To get the area of the isosceles triangle, we can simply multiply this formula by 2
We also have a sector that is divided in half. If we choose the angle
of half of the sector to be theta, we have that area equal to
Multiplying by 2, we find the area of the larger sector. In this case we have
To derive a formula for finding the area in terms of r and h, we can simply subtract the area of the isosceles triangle from the area of the entire sector. We have the following formula:
A final step is to write theta in terms of h and r. Using the right triangle,
we can use the trigonometric functions to do this. We know that 
By substitution, we get: 