Trapezoid, Quadrilateral Problems, 3

Category: Plane Geometry

"Published in Newark, California, USA"

Find the area of the rectilinear figure shown, if it is the difference between two isosceles trapezoids whose corresponding sides area parallel.

Photo by Math Principles in Everyday Life

Solution:

The given plane figure consists of the difference of two isosceles trapezoids. Let's analyze and label further the above figure as follows

Photo by Math Principles in Everyday Life

The area of a large trapezoid is

Before we solve for the area of a small trapezoid, we need to solve for the variables first.

Using Pythagorean Theorem

The height of a small trapezoid is

Since the given two trapezoids are isosceles trapezoid with the common thickness which is 2", then we can solve for the variables by using similar triangles. 

Using similar triangles

The length of the upper base of a small trapezoid is

Using similar triangles

The length of the lower base of a small trapezoid is

Hence, the area of a small trapezoid is

Therefore, the area of a plane figure is

Trapezoid - Circular Segment Problems, 2

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 isosceles triangle because the two base angles and their opposite sides are  equal. If ∆AOB is an isosceles triangle, then the altitude h bisects AB into two equal parts which are 1.5'' each.

The radius of a circular arc is

The height of ∆AOB is

The angle of a circular arc is

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