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

Area - Triangle, Given Three Vertices, 4

Category: Analytic Geometry, Plane Geometry

"Published in Newark, California, USA"

If the area of a triangle with vertices (5, 2), (x, 4), and (0, -3) is 12 ½, find x.

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 triangle is given by the formula

Substitute the values of the coordinates of the vertices as well as the area, we have

Therefore, x = 2 in which the other vertex of a triangle is (2, 4).

Square, Rectangle, Parallelogram Problems, 4

Category: Analytic Geometry, Plane Geometry

"Published in Newark, California, USA"

Show that the points (-1, 1), (0, -3), (5, 2), and (4, 6) are the vertices of a parallelogram, and find its area.

Solution:

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

Photo by Math Principles in Everyday Life

The first thing that we need to do is to get the slope of each sides of a parallelogram using the two point formula as follows

For the slope of AB:

For the slope of BC: 

For the slope of CD:

For the slope of AD:

Since

then the given four points are the vertices of a parallelogram. If the slope of two lines are equal, then the two lines are parallel. Parallelogram is a closed plane figure or quadrilateral whose two opposite sides are parallel. 

The area of a parallelogram is given by the formula

Substitute the values of the vertices of a parallelogram in order and solve for the value of a matrix, we have

 

 

 

 

Therefore, the area of a parallelogram is