Formula for calculating the diagonal of a rhombus

A rhombus is a quadrilateral with all sides equal and is a parallelogram with two diagonals perpendicular to each other, two opposite sides equal, and the diagonals bisecting each other at their point of intersection simultaneously bisecting each angle. The properties, definition, and identification criteria of a rhombus are extensively covered on Wikipedia, you can further explore on Wikipedia the article about rhombus to understand more. 

Method to compute the diagonal of a rhombus

Formula for calculating the diagonal of a rhombus

What is the diagonal of a rhombus? The diagonal of a rhombus is the line connecting the opposite vertices of the rhombus. The two diagonals of the rhombus are perpendicular to each other at their point of intersection. Diagonals are the most crucial quantity for determining the area of a rhombus.

The formula for calculating the area of a rhombus is: 

Area = 1/2 x d1 x d2

Where: d1, d2 are the diagonals of the rhombus.

Therefore: the formula to calculate the diagonal of a rhombus is d1 = 2S: d2

Special case:

Suppose we need to calculate the length of the diagonal of rhombus ABCD with side a and an angle ABC = 60 degrees -> how do we calculate the diagonal of the rhombus in this case?

Solution:

Since ABCD is a rhombus, all sides are equal to a.

Considering triangle ABC: AB = BC = a

Also, ABC = 60 degrees => Triangle ABC is an equilateral triangle with side length a.

=> AB = AC = BC = a

=> The length of the diagonal of a rhombus is AC = BD = a.

As there are many ways to solve this problem using different methods, but the solution above is one of the simplest and easiest formulas to understand.

2. Exercise: Calculate the diagonal of a parallelogram

Related to problems involving the calculation of the diagonal of a rhombus, please refer to Some examples of problems requiring students to calculate the diagonal of a rhombus. In these problems, you and students can rely on the formula for calculating the diagonal in the solution above to perform calculations.

Problem 1 - The problem states the length of one diagonal and asks students to find the length of the other diagonal.

a) A rhombus has a long diagonal of 9 cm, and the length of the short diagonal is 5/9 of the length of the long diagonal. Calculate the length of the short diagonal?

b) A rhombus has a difference in length between its two diagonals of 15 cm, and the first diagonal is 4 times the length of the second diagonal. Calculate the lengths of the two diagonals?

Solution: 

Let the long and short diagonals of the rhombus be d1 and d2 respectively.
a. d1 = 9cm = > d2 = 5/9.d1 = 5cm.
b. d1 - d2 = 15cm, d1 = 4d2 
Hence: 4d2  - d2 = 15 so d2 = 5cm, d1 = 20cm.

Problem 2 - The problem provides the perimeter or area of the rhombus, calculate the length of the diagonal.

A rhombus has an area of 5/3 m2, knowing the length of one diagonal is 25/2 dm. Calculate the length of the other diagonal?

Solution:

Let the diagonal of the rhombus be d1 = 25/2dm =  1.25m and d2.
The area of the rhombus is S = 1/2.d1.d2 = 1/2.1.25.d2 = 0.625d2 
=> d2 = S: 0.625 = 5/3 : 0.625 = 1.041m2.

Problem 3 - The problem provides the side and the height of the rhombus, requires students to find the lengths of the two diagonals.

Given rhombus ABCD with a side of 12.5cm, a height of 6.72 cm, and AC shorter than BD. What are the lengths of the two diagonals AC and BD respectively?

Solution:

Applying the formula to calculate the area of the rhombus: S = h.a = 6.72 x 12.5 = 84cm.
=> 1/2 AC x BD = 84 => 2AC.BD = 336

Let O be the intersection of the two diagonals of the rhombus.
We have AOB as a right triangle at O so AB² = OA² + OB² 

Where OA = 1/2 AC, OB = 1/2 BD
=> 12.5² = 1/4 (AC² + BD²) <=> 625 = AC² + BD²

AC² + BD² = 625 <=> AC² + BD²+ 2AC.BD = 625 + 336 <=> (AC + BD)² = 961 <=> AC + BD = 31 (1)

AC² + BD² = 625 <=> AC² + BD²- AC.BD = 625 -336 <=> (BD - AC)² = 289 <=> BD - AC = 17 (Given that BD > AC) (2)

From (1) and (2), we have:

BD = 24, AC = 7cm.

Above is a sharing about Formula for calculating the diagonal of a rhombus quite simple and basic because this knowledge is all in the curriculum of Mathematics in grade 1 and grade 2. You can also refer to how to calculate, the formula for calculating the area of a rhombus as well as the perimeter that Mytour has shared to review the knowledge before instructing, guiding back to your children.

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