Square, Rectangle, Parallelogram Problems, 9

Category: Plane Geometry

"Published in Newark, California, USA"

A certain city block is in the form of a parallelogram. Two of its sides are each 421 ft. long; the other two sides are each 227 ft. in length. If the distance between the first pair of sides is 126 ft., find the area of the land in the block, and the length of the diagonals.

Solution:

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

Photo by Math Principles in Everyday Life

Since the height of a parallelogram is given as well as the opposite sides, then we can solve for the area as follows

In order to solve for the diagonals of a parallelogram, we need to label further the figure above as follows

Photo by Math Principles in Everyday Life

Use Pythagorean Theorem for ∆ADE, we have

Use Pythagorean Theorem for ∆DEB, we have

Photo by Math Principles in Everyday Life

Use Pythagorean Theorem for ∆ACF, we have

Concentric Circles Problems

Category: Plane Geometry

"Published in Newark, California, USA"

A metal washer 1 in. in diameter is pierced by a ½ in hole. What is the area of one face of the washer?

Solution:

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

Photo by Math Principles in Everyday Life

A metal washer is made from two concentric circles and its a ring shape. 

Let d₁ = be the diameter of outer circle
      d₂ = be the diameter of inner circle
      A₁ = be the area of large circle
      A₂ = be the area of small circle
 
Area of Large Circle:
 

 

 

 
Area of Small Circle:
 

 

 

 

 
Therefore, the area of a metal washer is
 

 

 

 

 

or

 

 

Square, Rectangle, Parallelogram Problems, 8

Category: Plane Geometry

"Published in Vacaville, California, USA"

The official distance between home plate and second base in a baseball diamond is 120 ft. Find the area of the official ball diamond and the distances between the bases. (The official ball diamond is in the form of a square.)

Solution:

To illustrate the problem, it is better to post and label the picture as follows

Photo by Math Principles in Everyday Life

In this problem, the diagonal of a square is given which is a distance between the home plate and the second base. Since the official ball diamond is a square, then the diagonal bisects the square of side x into two equal isosceles right triangles.

Let x = be the distances between the bases
      d = be the distance between home plate and 2nd base
      A = be the area of official ball diamond 

Apply Pythagorean Theorem in order to solve for the distances between the bases as follows

or

Finally, the area of official ball diamond is