Fine Point in 4 Sided Polygons:

 

While tutoring students I felt like the students understood the various definitions of 4 sided polygons but they were missing the interrelationship between them.

 

Let me begin with various fundamental definitions. When I say fundamental it implies that nothing can be added or removed from the definitions without breaking the rules of mathematics or adding unnecessary matter.

 

Parallelogram

-          4 sides

-          Opposite sides are parallel

 

Rhombus

-          4 sides

-          Opposite sides are parallel and equal

 

Rectangle

-          4 sides

-          Opposite sides are parallel

-          All angles are 90°

 

Square

-          4 sides

-          Opposite sides are parallel and equal

-          All angles are 90°

 

Now if you look at the definitions and progress from parallelogram to square you will notice that we are adding restrictions to the definition of the parallelogram.

 

Thus, I can modify the original definition of the four polygons above as:

 

Parallelogram

-          4 sides

-          Opposite sides are parallel

 

Rhombus

-          Parallelogram with equal sides

 

Rectangle

-          Parallelogram with angles of 90°

 

Square

-          Rectangle with all sides equal

 

2^nd definition of a square

-          Rhombus with one angle 90°

 

The implication of the above is as follows:

-          All equations applicable to the parallelogram are also applicable to the rhombus, square, and rectangle

-          All equations applicable to the rectangle are applicable to a square

-          All equations applicable to a rhombus are applicable to a square

 

Example

 

The area of a rhombus is given as:

 

A =  1/2 x d₁ x d₂

 

-          here d1 and d2 are diagonals of a rhombus

 

Hence, the area of a square is:

 

A =  1/2 x d₁ x d₂

 

-          here d1 and d2 are diagonals of a square which happen to be equal

 

Example Problem

 

Q. What is the area of a square with a diagonal length of 40 cm?

 

A.

 

Since a square is also a rhombus, we can use the equation described above to find the area of the square.

 

Thus, the area of the square is:

 

A = 1/2x(40cm)x(40cm) = 800cm²

 

 

The benefit of the above understanding is that you do not have to go through the additional step of finding the side of the square first and then determining the area from the classic equation of the area of a square.