Rhombus

"What is a rhombus?" asked Miss Granger in geometry class.

Nicole, who had always loved geometry said, "A rhombus is also a parallelogram, but because of its unique properties, it gets an individual identity as a quadrilateral."

When you ask someone the question: 'What is a rhombus?,' then Nicole's answer is almost perfect.

In this section, we’re going to look at the rhombus definition, understand the rhombus shape, learn about the area of a rhombus, as well as its perimeter. But before we get into those details, let's also look at properties of a rhombus, and its unique diamond geometric shape. 

Lesson Plan

 

What is a Rhombus?

The rhombus is an interesting quadrilateral because of its unique shape.

While all its sides are equal, it is not a square because all its angles are \(90^\circ \).

In this chapter, we will explain the properties of a rhombus and why it's quite a cool quadrilateral.

The baseball field is diamond shaped and people usually compare it to a rhombus

Rhombus-shaped jewelry is popular

Like most things in math, the name "rhombus" comes from Ancient Greece from the word "rhombos," which means a piece of wood whirled on a string to make a roaring noise. Today, however, we see rhombus-shaped figures every day. Look at the diamond in a baseball field or those cool earrings you spotted at the mall the other day.

Rhombus definition: A rhombus can be defined as a diamond-shaped quadrilateral that has all four sides equal. But isn’t that a square? Yes, as a matter of fact, it is.

But how is a rhombus different? Well, unlike a square where all sides and angles are equal, the rhombus shape may show that it has all four sides equal, but only its opposite angles are equal.


Properties of a Rhombus

A rhombus, like all geometric shapes, has properties that are unique to it. These include:

  • All four sides of a rhombus are equal.
  • Opposite sides of a rhombus are parallel.
  • Opposite angles of a rhombus are equal.
  • Diagonals bisect each other at 90° or right angles.
  • The sum of two adjacent angles is 180°.
  • All rhombuses are parallelograms, but all parallelograms are not rhombuses.
  • All rhombuses are not squares, but all squares are rhombuses.

Formulae for a Rhombus

The formulae for rhombuses are defined for two features:

  1. Area of a rhombus, where \(A = \dfrac{{d_1 d_2 }}{2}\)
  2. Perimeter of a rhombus, where \(P = 4\times \text{side}\)

Area and Perimeter of a Rhombus

Area of a Rhombus

The area of a rhombus is defined as a product of the two diagonals divided by 2

\(A = \dfrac{{d_1 d_2 }}{2}\)

Where \(d_1\) and \(d_2\) are the lengths of the diagonals.

Perimeter of a Rhombus

Since all four sides of a rhombus are equal, much like a square, the formula for the perimeter is the product of the length of one side with 4

 \(P = 4 \times \text{side}\)


Angles of a Rhombus

As mentioned in the properties of a rhombus, the sum of two adjacent angles is 180°.

So if one of the angles is 60°, then the adjacent angle will be 120°.

Also, opposite angles of a rhombus are equal, which means summing up the four angles will give you 360°, i.e. the number of degrees that can be measured in any quadrilateral.


Diagonals of a Rhombus

One thing you should remember about the diagonal of a rhombus is that it always bisects at a 90° angle, which means that the two sides bisected will be the same length.

For example: if the length of a diagonal is 10 cm and the other diagonal bisects it, then it is divided into two 5 cm diagonals.

Unless the rhombus is a square, then the diagonals will have different values.

If you know the side of the rhombus and the value of certain angles, then you can determine the length of the diagonal.

 
important notes to remember
Important Notes
  1. Adjacent angles in a rhombus add up to 180°.
  2. Diagonals bisect each other at right angles.
  3. All squares are rhombuses, but not all rhombuses are squares.

Solved Examples on Rhombus

Example 1

 

 

Rajeev has drawn a rhombus where the lengths of the two diagonal lengths \(d_1\) and \(d_2\) of a rhombus are 5 cm and 10 cm, respectively. He asks his sister Lola to help him find the area. Can you help Lola find the answer?

Solution

Given:
Diagonal \(d_1 = 5 \text{ cm}\)
Diagonal \(d_2= 10 \text{ cm}\)

 \(A = \dfrac{{d_1 d_2 }}{2}\)

\(\begin{align}A &= \frac{{5\times 10}}{2} \\ A &= 25 \text{ sq. cm} \end {align} \)

\(\therefore\) \(\text{Area} = 25 \text{ sq. cm}\)
Example 2

 

 

Rahul was given the area of a rhombus and the length of one diagonal. The area, she was told, was 100 square cm and the length of a diagonal was 20 cm. Can you help Rahul find the length of the other diagonal?

Solution

Given:

\(Area  = 100 \text{ cm}^2 \text{ and } d_1 = 20 \text{ cm}\)

\(\begin{align}A &= \dfrac{{d_1\times d_2}}{2} \\ 100 &= \dfrac{20 \times {d_2}}{2} \\ d_2 &=10 \text{ cm} \end {align} \)

\(\therefore\) \(d_2 = 10 \text { cm}\)
Example 3

 

 

Sonia and Aman were playing a game of hopscotch when they spotted a rhombus-shaped tile at the playground. Somebody had inscribed 15 cm along the length and the width of the tile. Can you help Sonia and Aman find the perimeter of the tile?

Solution

Given:

Length of side \(= 15 \text{ cm}\)

Since all sides of a rhombus are equal, all four sides are equal to 15 cm.

\(Perimeter = 4 \times \text{Side}=4 \times 15=60\)

\(\therefore\) \(\text{Perimeter} = 60 \text{ cm}\)
Example 4

 

 

The size of the acute angle of a rhombus is half the size of its obtuse angle. Find the value of the two angles.

Solution

Given:

We know that the obtuse angle is twice the size of the acute angle.

We also know from the properties of a rhombus that the sum of two adjacent angles is 180°.

So let the acute angle be \(x\) and the obtuse angle be \(2x\)

\(\begin{align} x + 2x &= 180 \\ 3x &= 180\\ x &= 180\div3\\ x &=60\\ \therefore  2x &= 120 \end{align}\)

\(\therefore\) 1st angle \(= 60^\circ \) and 2nd angle \(= 120^\circ \)
Example 5

 

 

Aarav's house had a tiny rhombus-shaped storeroom with a perimeter of 120 m. If Aarav walked from one corner of the room to another, the length was 40 m. However, this wasn't the case when he walked the length of the other two corners. Can you help him find the length of the other diagonal so that he can find the area of her storeroom?

Solution

Given:

Let us divide this problem into three parts

Part 1

As we know, the perimeter is the sum of all four sides.

In a rhombus, as all four sides are equal,

\[P = 4\times \text{Side}\]

\[120 =  4\times \text{Side}\]

\[\therefore \text{Side} = 30 \text{ m}\]

Part 2

Now, we know the side, and we know the length of a diagonal. Since the diagonals bisect each other at right angles, we know that the 40 m long diagonal is bisected. Using Pythagoras’ theorem, we can find the length of the other diagonal.

\(\begin {align} BC^2 &= OB^2 + OC^2\\ 30^2 &= 20^2 + OB^2\\ 900 &= 400 + OB^2\\ 900-400 &= OB^2\\ OB^2 &= 500\\ \therefore OB &= \sqrt{500}\\ OB &= \sqrt{ 100\times5}\\ OB &= 10\sqrt{5} \end {align}\)

\(\begin {align} OB + OD &= BD\\ OB&=OD\\ \therefore 2OB &=BD\\BD &= 2 \times 10 \sqrt{5}\\BD &=20\sqrt{5}\\ \end{align}\)

Part 3

Now, let us calculate the area of the rhombus now that we have the length of both diagonals.

\(\begin{align}A &= \dfrac{{d1\times d2}}{2} \\ A &= 20 \sqrt{5} \times {40}\div {2} \\ A &=400 \sqrt{5} \text{ sq. m} \end {align} \)

\(\therefore\) \(A = 400\sqrt5 \text{ sq. m}\)

Interactive Questions on Rhombus

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 
 
tips and tricks
Tips and Tricks
  1. A parallelogram is a rhombus if its diagonals are perpendicular to each other.
  2. The two diagonals aren’t equal in length. If they are, then the rhombus is a square.

Let's Summarize

We hope you enjoyed learning about rhombus with the solved examples and interactive questions. Now you will be able to easily solve questions related to the rhombus.

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Frequently Asked Questions (FAQs)

1. Are all squares rhombuses?

Yes, all squares are rhombuses. In fact, you may even call it a very special case of a rhombus. Like all rhombuses, all four sides of the square are equal. 

2. What shape is a rhombus?

A rhombus is a quadrilateral that is diamond geometric shaped. People say that if you look at a pack of cards and pull out the 13 diamond cards, the diamond geometric shape is that of a rhombus.

3. Are all rhombuses rectangles?

You may argue that all squares are rhombuses and all squares are also rectangles, but does that mean that all rhombuses are rectangles? If you said yes, then you’d have to look at some really special conditions. Firstly, remember that all four sides of the rhombus are equal. This isn’t the case with a rectangle. A rhombus will become a rectangle only if its properties are identical to a square. This is because all squares are rhombuses.   

4. What is the area of a rhombus?

The area of a rhombus is calculated by dividing the product of the diagonals by 2. Mathematically, this can be defined as:
\(A = \frac{{d_1 d_2 }}{2}\)
Where d = is the length of each diagonal

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