Rhombus-01

Rhombus

Definition of Rhombus

 

A rhombus is a four-sided quadrilateral. In a rhombus, opposite sides are parallel and the opposite angles are equal. Additionally, all the sides of a rhombus are equal in length, and the diagonals bisect each other at right angles. A rhombus is a unique case of a parallelogram. The rhombus is also called a diamond or rhombus diamond. The plural form of a rhombus is rhombi or rhombuses.

In other words, 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. This is why it is can also be called a tilted square.

It is important to note that every rhombus is 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. 

Thus, we can conclude that:

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

A rhombus has three additional names: Diamond, Lozenge, Rhomb

RHOMBUS IN EVERY DAY

A rhombus is a diamond-shaped quadrilateral that has all four sides as equal and diagonals are perpendicular bisectors. We see rhombus-shaped figures in our day-to-day lives. Some of the real-life examples of a rhombus are kite, a diamond, an earring, etc.

PROPERTIES OF ANGLES OF RHOMBUS

Listed below are some important properties of the angles of a rhombus:

 

  1. The rhombus has four interior angles.
  2. The sum of interior angles of a rhombus adds up to 360 degrees.
  3. The opposite angles of a rhombus are equal to each other. 
  4. In a rhombus, diagonals bisect each other at right angles.
  5. The adjacent angles are supplementary.
  6. The diagonals of a rhombus bisect these angles.

 AREA OF RHOMBUS

The area of the rhombus is the region covered by it in the two-dimensional plane. 

The formula for the area of a rhombus is equal to the product of diagonals of rhombus divided by 2. It can be represented as:

Area of Rhombus, A = (d1 x d2)/2 square units

where d1 and d2 are the diagonals of a rhombus.

PERIMETER OF RHOMBUS

The perimeter of a rhombus is the total length of its boundaries. Or the sum of all the four sides of a rhombus is its perimeter. The formula for its perimeter is given by:

Perimeter of Rhombus, P = 4a units

Where the diagonals of the rhombus are d1 & d2 and ‘a’ is the side.

PROPERTIES OF RHOMBUS

Some of the important properties of the rhombus are listed below:

  • All sides of the rhombus are equal.
  • The opposite sides of a rhombus are parallel.
  • Opposite angles of a rhombus are equal.
  • In a rhombus, diagonals bisect each other at right angles.
  • Diagonals bisect the angles of a rhombus.
  • The sum of two adjacent angles is equal to 180 degrees.
  • The two diagonals of a rhombus can be divided into four right-angled triangles which are congruent to each other
  • We get a rectangle when you join the midpoint of the sides.
  • We will get another rhombus when you join the midpoints of half of the diagonal.
  • Around a rhombus, there can be no circumscribing circle that can be prescribed
  • Within a rhombus, there can be no inscribing circle that can be prescribed
  • The rectangle that we get when we join the midpoints of the 4 sides, the length and width of the rectangle will be half the value of the main diagonal so that the area of the rectangle will be half of the rhombus.
  • When the shorter diagonal is equal to one of the sides of a rhombus, two congruent equilateral triangles are formed.
  • We will get a cylindrical surface having a convex cone at one end and concave cone at another end when the rhombus is revolved about any side as the axis of rotation.
  • We will get a cylindrical surface having concave cones on both the ends when the rhombus is revolved about the line joining the midpoints of the opposite sides as the axis of rotation.
  • We will get a solid with two cones attached to their bases when the rhombus is revolved about the longer diagonal as the axis of rotation. In this case, the maximum diameter of the solid is equal to the shorter diagonal of the rhombus.
  • We will get solid with two cones attached to their bases when the rhombus is revolved about the shorter diagonal as the axis of rotation. In this case, the maximum diameter of the solid is equal to the longer diagonal of the rhombus.

Videos (3)

Examples

Q1 .The two diagonal lengths d1 and d2 of a rhombus are 8cm and 12 cm, respectively. Find its area.

Solution:

Given:

Diagonal d1 = 8cm

Diagonal d2= 12 cm

Area of the rhombus, A = (d1 x d2)/2 square units

A = (8 x 12)/2

A = 96/2

A = 48 cm2

Therefore, the area of rhombus = 48 cm2

Q2. Savant has drawn a rhombus where the lengths of the two diagonals d1 and d2 are 15 units and 10 units, respectively. He asks his sister Lalita to help him find the area. Can you help Lalita find the answer?

Solution:

Given:

Diagonal, d1 = 15 units, and d2 = 10 units

A = (d1 × d2)/2

A = (15×10)/2

A = 75 sq. units

Answer: The area of the rhombus = 75 sq. units.

Q3.: Find the diagonal of a rhombus if its area is 120 cm2 and length measure of longest diagonal is 20 cm.

Solution:

Given: Area of rhombus = 120 cm2 and Lets say d1 = 20 cm.

Using Area of the rhombus formula,  

A = (d1 x d2)/2 square units, 

we get

120 = (20 x d2)/2

120 = 10 x d2

or  12 =  d2

Therefore, the Length of another diagonal is 12 cm.

Q4 .List four basic properties of a rhombus?

Solution:

The basic properties of the rhombus are:

  1. The opposite angles are congruent.
  2. The diagonals intersect each other at 90 degrees.
  3. The diagonals bisect the opposite interior angles.
  4. The adjacent angles are supplementary.

Q5.Ram was given the area of a rhombus, 200 square units, and the length of one diagonal as 20 units. Can you help Ram find the length of the other diagonal? Solution:

Solution:

Area = 200 sq. units, and diagonal d1 = 20 units

A = (d1 × d2)/2

200 = (20 × d2)/ 2

d2 = 20 units

Answer: The length of the other diagonal is 20 units.

Q6. What is the perimeter of a rhombus whose sides are all equal to 8 cm?

Solution:

Given, the side of rhombus = 8cm

Since all the sides are equal, therefore,

Perimeter = 4 x side

P = 4 x 8

P = 32 cm



Q7. Ram and Shyam 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 12 units. Can you help Ram and Shyam find the perimeter of the tile?

Solution:

Length of the tile = 12 units.

Since all sides of a rhombus are equal, all four sides are equal to 12 units.

Perimeter = 4 × side = 4 × 12 = 48 units

Answer: The perimeter of the tile = 48 units.

Practice Questions

Rhombus Practice Questions

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A rhombus is also known as:

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How many equal sides does a rhombus has?

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The shape of a rhombus is:

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The opposite sides of a rhombus are:

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How many interior angles are there in a rhombus?

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Related Study Materials

FAQ

Is a rhombus a square?

No, rhombus is not a square but a square is a rhombus.

Why is a rhombus not a square?

Rhombus is not a square since for a square all the sides are equal and all the interior angles are equal, right angles. However, in rhombus all the interior angles are not equal even though they have equal sides.

Rational numbers are terminating decimals or non terminating and repeating whereas irrational numbers are non-terminating and non-recurring. 

Example of the rational number is 1/2, and an irrational number is Pi(π) which is equal to 3.141592653589…….

Does a rhombus have 4 right angles?

No, a rhombus does not have four right angles.

Are all angles of a rhombus equal?

No, in rhombus only the opposite angles are equal.

Does a rhombus add up to 360?

We know that the sum of all the interior angles of a quadrilateral is equal to 360 degrees. Hence, the angles of a rhombus add up to 360 degrees.

Is Square a Rhombus?

Rhombus has all its sides equal and so does a square. Also, the diagonals of the square are perpendicular to each other and bisect the opposite angles. Therefore, a square is a type of rhombus.

What is unique about a rhombus?

One of the two characteristics that make a rhombus unique is that its opposite sides are parallel. The other identifying property is that its four sides are equal in length, or congruent.

How did the term rhombus get coined?

A rhombus can be a square, or if it has two acute angles and two obtuse angles, it takes the shape of a diamond. The rhombus gets its name from the Greek word rhómbos, which means "a spinning top." This word describes the shape of a "top," an object that was tied to a cord and spun around, making a great noise and looks like the geometric figure rhombus.

For example, 2/3 + 6/3 = 8/3.

What is altitude of a rhombus?

The height of a rhombus, also called its altitude, is the shortest perpendicular distance from its base to its opposite side.

What is the base of rhombus?

The base of the rhombus is one of its four sides, and the height is the altitude which is the perpendicular distance from the chosen base to the opposite side.

Do diagonals of a rhombus bisect each other?

The opposite angles of a rhombus are equal. The diagonals of a rhombus bisect each vertex angle. Additionally, the diagonals of a rhombus bisect each other at right angles

   
Acute Angled TriangleRight Angled TriangleObtuse Angled Triangle

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