## Sunday, April 20, 2014

### Trapezoid - Circular Segment Problems

Category: Plane Geometry, Trigonometry

"Published in Vacaville, California, USA"

The plane area shown in the figure consists of an isosceles trapezoid (non-parallel sides equal) and a segment of a circle. If the non-parallel sides are tangent to the segment at points A and B, find the area of the composite figure.

 Photo by Math Principles in Everyday Life

Solution:

The given plane figure consists of an isosceles trapezoid and a circular segment. Let's analyze and label further the above figure as follows

 Photo by Math Principles in Everyday Life

From point A, draw a line perpendicular to CA and from point B, draw a line perpendicular to BD. The intersection of the two lines is point O which is the center of a circular arc. By using the laws of angles, ∆AOB is an equilateral triangle because all interior angles and all sides are all equal. If ∆AOB is an equilateral triangle, then the altitude h bisects AB into two equal parts which are 1.5'' each.

By Pythagorean Theorem

Hence, the area of a circular segment is

The height of a circular segment is

The height of a trapezoid is

The length of the upper base of a trapezoid is

but

Then the above equation becomes

Hence, the area of a trapezoid is

Therefore, the area of a plane figure is