Area of Rhombus Shortcut

An edge of a Math enthusiasts than the normal the people is they can create their own formula. Last year while we were reviewing for elimination round of MMC, I came across with this question, “The diagonal of a rhombus differ by 4. If its perimeter is 40, find its area”.

This problem could be easy for everyone but not for beginners, let me show how to derive a shortcut formula for this problem.

Before we do that, we need first to know some important properties of Rhombus that will help us along the way.

  • Each side of a rhombus is equal
  • The intersection of the diagonals of rhombus forms a right angle

We will base our derivation using the figure below

rhombus

 

Let ABCD be a rhombus with side s and diagonals d_1  and  d_2. The diagonals intersect at point O.

Objective: Find a shortcut formula to solve for the area of the rhombus given a side s and the difference of the length of diagonals.

Area of Rhombus:

A=\displaystyle\frac{d_1d_2}{2}

Rearranging

2A=d_1d_2  (i)

Observe in the figure that \triangle AOD is a right triangle. By Pythagorean theorem, we have the following relations.

(\displaystyle\frac{d_1}{2})^2+(\displaystyle\frac{d_2}{2})^2=s^2

\displaystyle\frac{d_1^2}{4}+\displaystyle\frac{d_2^2}{4}=s^2

d_1^2+d_2^2=4s^2

Consider the identity (d_1-d_2)^2=d_1^2-2d_1d_2+d_2^2.

This can be written in d_1^2+d_2^2=(d_1-d_2)^2+2d_1d_2

By substitution we have,

d_1^2+d_2^2=4s^2

(d_1-d_2)^2+2d_1d_2=4s^2

From (i) we have 2A=d_1d_2

By another substitution,

(d_1-d_2)^2+2(2A)=4s^2

4A=4s^2-(d_1-d_2)^2

\boxed{A=s^2-(\displaystyle\frac{d_1-d_2}{2})^2}

What about if the given is the sum of the lengths of diagonal?

Then we have to consider the following identity

(d_1+d_2)^2=d_1^2+2d_1d_2+d_2^2

Do the same derivation technique until you get the following formula.

\boxed{A=(\displaystyle\frac{d_1+d_2}{2})^2-s^2}

Going back to the problem posted above, using the derived formula the answer must be 96 sq. units.

Dan

Dan

Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.
Dan

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