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# Properties of Rectangle – Angles, Sides & Diagonals

January 13, 2023

This post is also available in: हिन्दी (Hindi)

A rectangle is a closed 2D shape with four sides, four angles, and four vertices. It’s a type of quadrilateral and parallelogram whose measure of all angles is $90^{\circ}$. The word ‘rectangle’ starts with the Latin word ‘rectus’, which means ‘right’ or ‘straight’.

Let’s understand what are the properties of rectangle with examples.

## Properties of a Rectangle

The following are the most important properties of a rectangle.

• A rectangle is a quadrilateral.
• The opposite sides of a rectangle are equal and parallel to each other.
• The interior angle of a rectangle at each vertex is $90^{\circ}$.
• The sum of all interior angles is $360^{\circ}$.
• The diagonals bisect each other.
• The length of the diagonals is equal.
• The length of the diagonal with sides a and b is, diagonal = $\sqrt{a^{2} + b^{2}}$
• Properties of a Rectangle
• The following are the most important properties of a rectangle.
• A rectangle is a quadrilateral.
• The opposite sides of a rectangle are equal and parallel to each other.
• The interior angle of a rectangle at each vertex is $90^{\circ}$.
• The sum of all interior angles is $360^{\circ}$.
• The diagonals bisect each other.
• The length of the diagonals is equal.
• The length of the diagonal with sides a and b is, diagonal = $\sqrt{a^{2} + b^{2}}$

## Diagonal of a Rectangle

The line segments that join the opposite corners of a rectangle are called diagonals.

In the above figure, the two diagonals of the rectangle $\text{ABCD}$ are $\text{AC}$ and $\text{BD}$. The diagonals of a rectangle are of the same length., i.e., in the rectangle $\text{ABCD}$, $\text{AC} = \text{BD}$.

If we know the length and width of a rectangle, we can calculate the diagonal of a rectangle using the Pythagorean Theorem.

In the above figure, $\angle \text{ADB}$ is right-angled at $\text{A}$. The diagonal $\left( \text{BD} \right)$ of the rectangle forms its hypotenuse.

So, using the Pythagorean theorem, we get, $\text{Diagonal}^2 = \text{Length}^2 + \text{Width}^2$

Therefore, $\text{Diagonal } = \sqrt{\text{Length}^2 + \text{Width}^2}$.

## The Length of the Diagonals of a Rectangle Is Equal

Consider a rectangle $\text{ABCD}$, with diagonals $\text{AC}$, and $\text{BD}$.

Now, consider $\triangle \text{ABC}$ and $\triangle \text{DCB}$.

$\angle \text{ABC} = \angle \text{DCB} = 90^{\circ}$ (Angles of rectangle)

$\text{BC} = \text{BC}$  (common side)

$\text{AB} = \text{DC}$  (Opposite sides of a parallelogram are equal)

Therefore, $\triangle \text{ABC} \cong \triangle \text{DCB}$ (SAS Congruence Criterion)

Thus, $\text{AC} = \text{BD}$  (Corresponding Parts of Congruent Triangles)

Hence the diagonals of a rectangle are equal.

## Diagonals of a Rectangle Bisect Each Other

Consider a rectangle $\text{ABCD}$ with diagonals $\text{AC}$ and $\text{BD}$ intersecting at $\text{O}$.

In $\triangle \text{OAD}$ and $\triangle \text{OCB}$,

$\angle \text{ODA} = \angle \text{OBC}$ (Alternate interior angles as $\text{AD} || \text{BC}$ and $\text{BD}$ is a transversal)

$\text{AD} = \text{BC}$ (Opposite sides of a rectangle are equal)

$\angle \text{OAD} = \angle \text{OCB}$ (Alternate interior angles as $\text{AD} || \text{BC}$ and $\text{AC}$ as transversal)

Therefore, $\triangle \text{OAD} \cong \triangle \text{OCB}$  (ASA Congruence Criterion)

Equating the corresponding parts of congruent triangles, we get:

Hence, $\text{AO} = \text{CO}$ and $\text{BO} = \text{DO}$ (Corresponding Parts of Congruent Triangles)

Thus, the diagonals of a rectangle bisect each other.

## What is a Rectangle?

A rectangle is a closed 2D shape, having four sides, four angles, and four vertices. Each of the four angles of a rectangle is a right angle, i.e., measure $90^{\circ}$. The opposite sides of a rectangle are equal and parallel. Since a rectangle is a 2D shape, it is characterized by two dimensions, length, and width. Length is the longer side of the rectangle and width is the shorter side.

In the above figure, $\text{ABCD}$ is a rectangle. The longer sides $\text{AB}$ and $\text{CD}$ are called length and the shorter sides $\text{BC}$ and $\text{AD}$ are called width(or breadth). The four angles of the rectangle $\text{ABCD}$ are $\angle \text{A}$, $\angle \text{B}$, $\angle \text{C}$, and $\angle \text{D}$ each equal to $90^{\circ}$.

## Examples of Rectangle

A rectangle is the most common shape that we see around ourselves in our daily lives. Some of the common examples of rectangular shapes are table top, blackboard, notebook, door, window, envelope, drawer, etc.

## Other Names of Rectangle

Rectangles are a special case of parallelograms. These are the other names of a rectangle.

• A rectangle is also called an equiangular quadrilateral since all the angles of a rectangle are equal.
• It is also called a right-angled parallelogram since a rectangle has parallel sides.

## Practice Problems

1. What is a rectangle?
2. Under what condition a parallelogram is called a rectangle?
3. Under what condition a rectangle is called a square?
4. State True or False
• Adjacent sides of a rectangle are equal
• Opposite sides of a rectangle are equal
• Adjacent sides of a rectangle are parallel
• Adjacent sides of a rectangle are perpendicular
• Opposite sides of a rectangle are parallel
• Opposite sides of a rectangle are perpendicular
• Diagonals of a rectangle are equal
• Diagonals of a rectangle are unequal
• Diagonals of a rectangle are perpendicular to each other
• Diagonals of a rectangle are parallel to each other

## FAQs

### What is a rectangle in math?

A rectangle is a closed 2D shape, having four sides, four angles, and four vertices. Each of the four angles of a rectangle is a right angle, i.e., measure $90^{\circ}$. The opposite sides of a rectangle are equal and parallel. Since a rectangle is a 2D shape, it is characterized by two dimensions, length, and width. Length is the longer side of the rectangle and width is the shorter side.

### What are the 4 properties of a rectangle?

The four important properties of a rectangle are
a) The opposite sides of a rectangle are equal and parallel to each other.
b) The interior angle of a rectangle at each vertex is $90^{\circ}$.
c) The diagonals bisect each other.
d) The length of the diagonals is equal.

### Is a rectangle a square?

A square is a special case of a rectangle. In other words, a rectangle is sometimes a square, when its adjacent sides are also equal.

### Why is it called a rectangle?

The word ‘rectangle’ starts with the Latin word ‘rectus’, which means ‘right’ or ‘straight’. Because of its right angles, a rectangle has straight sides.

### Why is a rectangle not a regular polygon?

A regular polygon is a polygon in which all the sides are of equal length and all the angles are of equal measure. In a rectangle, only the opposite sides are of equal length, therefore, a rectangle is not a regular polygon.

## Conclusion

A rectangle is a closed 2D shape, having four sides, four angles, and four vertices. Each of the four angles of a rectangle is a right angle, i.e., measure $90^{\circ}$. The opposite sides of a rectangle are equal and parallel. Since a rectangle is a 2D shape, it is characterized by two dimensions, a longer one is length, and a shorter one is width.