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

Latest posts by Dan (see all)

You may also like...

2 Responses

  1. Eldridge says:

    Thanks for sharing superb informations. Your web site is very cool. I’m impressed by the details that you’ve on this site. It reveals how nicely you understand this subject. Bookmarked this website page, will come back for extra articles. You, my friend, ROCK! I found simply the info I already searched all over the place and simply couldn’t come across. What a great web-site.

  2. I am really enjoying the theme/design of your blog. Do you ever run into any web browser compatibility issues? A number of my blog audience have complained about my blog not operating correctly in Explorer but looks great in Chrome. Do you have any tips to help fix this issue?

Leave a Reply

Your email address will not be published. Required fields are marked *