Two Column Proofs – Proving a Parallelogram Is a Rhombus – Geometry
Two Column Proofs – Proving a Parallelogram Is a Rhombus – Geometry

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Rhombus

A rhombus is a closed two-dimensional plane figure. It is considered a special parallelogram, and because of its unique properties, it gets an individual identity as a quadrilateral. A rhombus is also called an equilateral quadrilateral since all of its sides are equal in length. The term ‘rhombus’ has been derived from the ancient Greek word ‘rhombos’, which actually means something that spins. Let us learn more about the rhombus and its properties in this article.

 1 What is a Rhombus? 2 Properties of a Rhombus 3 What does a Rhombus look like? 4 Rhombus Formulas 5 FAQs on Rhombus

## What is a Rhombus?

A rhombus can be defined as a special parallelogram as it fulfills the requirements of a parallelogram, i.e. a quadrilateral with two pairs of parallel sides. In addition to this, a rhombus has all four sides equal just like a square. That is why it is also known as a tilted square. Look at the image below to understand the relationship of rhombus shape with parallelogram and square.

In the above figure, we can notice that every rhombus we see will also be a parallelogram, but not every parallelogram is a rhombus. A square can be considered as a special case of a rhombus because it has four equal sides. All the angles of a square are right angles, but the angles of a rhombus need not necessarily have to be right angles. And, hence a rhombus with right angles can be considered a square. Hence, we can conclude that:

• All rhombi or rhombuses are parallelograms, but all parallelograms are not rhombuses.
• All rhombi or rhombuses are not squares, but all squares are rhombuses.

A rhombus can have three additional names:

• Diamond
• Lozenge
• Rhomb

## Properties of a Rhombus

A rhombus is considered to be one of the special parallelograms as it has all the properties of a parallelogram. A rhombus has its two diagonals as its two lines of symmetry. Axis of symmetry can be considered as a line that divides an object into two equal halves. It creates a mirror-like reflection of both sides of the object. A rhombus is said to have reflection symmetry over both of its diagonals. The general properties of a parallelogram are as follows:

• Opposite angles are congruent or equal.
• The opposite sides are equal and parallel.
• Diagonals bisect each other.
• The sum of any two adjacent or consecutive angles is 180°.

A rhombus, like all geometric shapes, has properties that are unique to it. The table below describes the properties of a rhombus.

 Properties of Rhombus Reference All sides of a rhombus are congruent (equal). Here, AB = CD = DA = BC Diagonals bisect each other at 90° or we can also say that each of the two diagonals in a rhombus is the perpendicular bisector of the other. Here, diagonals DB and CA bisect each other at 90° Opposite angles are equal and the opposite sides are parallel. Here, CD || AB and BC || AD. ∠A = ∠C and ∠D = ∠B Adjacent angles add up to 180° ∠A + ∠B = 180° ∠B + ∠C = 180° ∠C + ∠D = 180° ∠A + ∠D = 180°

One thing we should remember about the diagonal of a rhombus is that in addition to bisecting each other at 90°, the two diagonals bisected will be of 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 segments. If you know the side of the rhombus and the value of certain angles, then you can determine the length of the diagonal.

## What Does a Rhombus Look Like?

A rhombus is a diamond-shaped quadrilateral that has all four sides equal. We can see rhombus-shaped figures in our day-to-day lives. Some of the real-life examples of a rhombus are shown in the below-given figure: a diamond, a kite, an earring, etc.

## Rhombus Formulas

The formulae for rhombuses are defined for two attributes:

1. Area of a rhombus, A = 1/2 × d1 × d2, where d1 and d2 are diagonals of a rhombus.
2. The perimeter of a rhombus, P = 4 × a, where a is the side.

### Area and Perimeter of a Rhombus

The area of a rhombus can be defined as the amount of space enclosed or encompassed by a rhombus in a two-dimensional plane. It is half of the product of the lengths of the diagonals. So, A = 1/2 × d1 × d2, where d1 and d2 are the lengths of the diagonals.

Just like a square, all four sides of a rhombus are equal, so, the formula for the perimeter of the rhombus is the product of the length of one side by 4. We get P = (4 × a) units, where a is the side of a rhombus.

☛ Important Notes on Rhombus

Here is a list of a few points that should be remembered while studying about rhombus:

• Adjacent angles in a rhombus are supplementary, they sum up to 180°.
• Diagonals bisect each other at right angles.
• All squares are rhombuses, but not all rhombuses are squares.

☛ Related Topics:

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## Rhombus Examples

1. Example 1: David has drawn a rhombus where the lengths of the two diagonals d1 and d2 are 5 units and 10 units, respectively. He asks his sister Linda to help him find the area. Can you help Linda find the answer?

Solution:

Given:

Diagonal, d1 = 5 units, and d2 = 10 units

A = (d1 × d2)/2

A = (5×10)/2

A = 25 sq. units

Answer: The area of the rhombus = 25 sq. units.

2. Example 2: Elsa was given the area of a rhombus, 100 square units, and the length of one diagonal as 20 units. Can you help Elsa find the length of the other diagonal?

Solution:

Given:

Area = 100 sq. units, and diagonal d1 = 20 units

A = (d1 × d2)/2

100 = (20 × d2)/ 2

d2 = 10 units

Answer: The length of the other diagonal is 10 units.

3. Example 3: Sam and Victor were playing a game of hopscotch and they spotted a rhombus-shaped tile at the playground. The length of each side of the tile was 15 units. Can you help Sam and Victor find the perimeter of the tile?

Solution:

Given:

Length of the tile = 15 units.

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

Perimeter = 4 × side = 4 × 15 = 60 units

Answer: The perimeter of the tile = 60 units.

## FAQs on Rhombus

### What is Rhombus in Geometry?

A rhombus is a 2-D shape with four sides hence termed as a quadrilateral. It has two diagonals that bisect each other at right angles. It also has opposite sides parallel and the sum of all the four interior angles is 360 degrees.

### Are All Squares Rhombuses?

Yes, all squares are rhombuses. A square can be considered as a special case of a rhombus because it has four equal-length sides. All the angles of a square are right angles, but the angles of a rhombus need not necessarily have to be right angles. Hence a rhombus with right angles can be considered a square.

### What Shape is a Rhombus?

A rhombus is a diamond-shaped quadrilateral. If you look at a pack of cards and pull out the 13 diamond cards, you will find that the diamond geometric shape is that of a rhombus. It has all four sides equal and opposite sides parallel to each other. It looks like a tilted square.

### Is a Rhombus a Regular Polygon?

No, a rhombus is not a regular polygon. A regular polygon must be equiangular (all of its angles are the same measure) and congruent or, equilateral (all of its sides are the same length). But a rhombus is only equilateral: all of its sides are of the same length and only the opposite angles are equal. A rhombus can never be considered a regular polygon as it is only an equilateral polygon and not an equiangular polygon.

### 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 d1 and d2 are the diagonals of a rhombus.

### What are the Four Properties of a Rhombus?

A rhombus is a quadrilateral that has the following four properties:

• Opposite angles are always equal.
• All sides are equal and opposite sides are parallel to each other.
• Diagonals bisect each other at a 90-degree angle and at equal lengths.
• The sum of any two adjacent angles will always be supplementary 180°.

### What is the Difference Between Rhombus and a Square?

The main difference between a rhombus and a square is that the angles of a rhombus are not equal to 90° whereas all the angles of a square are equal to 90°. It is often said that a square is a rhombus, but a rhombus isn’t always a square.

### What is the Difference Between a Parallelogram and a Rhombus?

The difference between a parallelogram and a rhombus is that in a rhombus all four sides are equal in measurement, whereas a parallelogram has opposite sides equal.

### What are the Basic Rhombus Formulas?

A few basic rhombus formulas are listed below:

• Area of the rhombus = 1/2 × d1 × d2 square units.
• The perimeter of a rhombus = (4 × side) units.

What is the Sum of the Interior Angles in a Rhombus?

The sum of the interior angles in a rhombus is 360°.

### Are All Angles of a Rhombus 90°?

No, the angles of a rhombus need not necessarily have to be right angles. A rhombus with all four interior angles as 90-degree angles is considered a square.

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