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

Circles - Circumference, Area Problems

Category: Plane Geometry

"Published in Vacaville, California, USA"

A storage bin of circular base has 324 sq. ft. of floor space. Find the radius of the floor.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

 

 

The area of a circular storage bin is given by the formula

Substitute the value of the area of a circular storage bin in order to solve for its radius, we have

Trapezoid, Quadrilateral Problems, 2

Category: Plane Geometry

"Published in Vacaville, California, USA"

 A sail has a spread of canvass as shown in the figure. Find the surface area of one side of the sail.

Photo by Math Principles in Everyday Life

Solution:

The given sail canvass is a general quadrilateral because the two opposite sides are not parallel. Because of this, we need to divide this into two triangles as follows

Photo by Math Principles in Everyday Life

The first triangle is a right triangle because the two adjacent sides are perpendicular to each other. Consider the right triangle at the left side. The area of a first triangle is

Consider the general triangle at the right side. In order to get the value of x, use Pythagorean Theorem at the first triangle, we have
 

 

 

 

 

 

   
Since the other side of a second triangle is also 15 ft., then the second triangle is an isosceles triangle. Use Heron's Formula in order to get the area of the second triangle, we have
 

   
where a, b, and c are the sides of a triangle and s is the semi-perimeter of a triangle.

The perimeter of a triangle is
 

 

 

   
The semi-perimeter of a triangle is
 

 

 

   
Hence, the area of a second triangle is
 

 

 

 

   
Therefore, the area of a sail canvass is
 

 

Trapezoid, Quadrilateral Problems

Category: Plane Geometry

"Published in Vacaville, California, USA"

The vertical end of a trough has the following dimensions: width at the top 4.4 ft., width at the bottom 3.2 ft., depth 3.5 ft. Find the area of the end of the trough.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

From the description of the given word problem, the vertical end of a trough is a trapezoid. The area of a trapezoid is given by the formula

Substitute the values of h, b₁ and b₂ to the above equation, we have