## Thursday, January 16, 2014

### Circular Segment Problems

Category: Plane Geometry, Trigonometry

"Published in Vacaville, California, USA"

Find the area of the shaded region in the figure: Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above Photo by Math Principles in Everyday Life

The area of a circular segment is the difference of the area of circular sector and a triangle.

The area of a circular sector is given by the formula

Substitute the values of r and θ, we have

Since the unit of area is square units, then the unit of an angle must be in radians since it is a unit less value of angle. Let's convert the unit of an angle in radians as follows

The area of a triangle given the two sides and the adjacent angle is given by the formula

Since the triangle is inscribed in a circle whose vertex is a center of a circle, then it follows that a = b = r. The above equation becomes

Substitute the values of r and θ, we have

Therefore, the area of circular segment is