# What is a Square in Geometry – (Definition, Shape, Properties & Examples)

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There are many types of quadrilaterals that you study such as rectangle, rhombus, kite, parallelogram, and trapezium. Square is one type of quadrilaterals. A square is a 2D shape with four equal sides and four equal angles.

Let’s understand what is a square in geometry and its properties with examples.

## What is a Square in Geometry?

A square is a type of parallelogram in which the adjacent sides are equal and the measure of angles is $90^{\circ}$. In other words, a square is a closed two-dimensional shape (2D shape) with all four sides equal.

In the above figure, $\text{ABCD}$ is a square, where $\text{AB} = \text{BC} = \text{CD} = \text{DA}$ and $\angle \text{A} = \angle \text{B} = \angle \text{C} = \angle \text{D} = 90^{\circ}$.

## Examples of Square

A square is a very common shape that we see around ourselves in our daily lives. Some of the common examples of square shapes are chess board, clock dial, picture frame, pizza box, coasters, keys on a keyboard, etc.

## Diagonal of a Square

The line segments that join the opposite vertices of a square are called diagonals.

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

If we know the length of an edge(or side) of a square, we can calculate the diagonal of a square 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 square forms its hypotenuse.

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

Therefore, $\text{Diagonal } = \text{Side } \times \sqrt{2}$.

## Properties of a Square

The following are the important properties of a square that helps you to identify it from the other quadrilaterals.

• All four sides of the square are equal to each other
• The opposite sides of a square are parallel to each other
• The interior angles of a square measure $90^{\circ}$
• The sum of all interior angles is $360^{\circ}$
• There are two diagonals in a square
• Each diagonal divide the square into two congruent triangles
• The length of the diagonals in a square is equal
• The diagonals of a square bisect each other at $90^{\circ}$
• Since the sides of a square are parallel, it is a type of a parallelogram

## Square vs Rectangle

The two quadrilaterals – square and rectangle have similar shapes and have some similarities as well as some differences. Let’s look at the similarities and differences between square and rectangle.

### Similarities Between Square and Rectangle

There are some properties that are common to a square and a rectangle. The following are the common properties in a square and rectangle.

• Both a square and a rectangle are quadrilaterals with four sides and four vertices
• The opposite sides of a square and a rectangle are parallel to each other
• The measure of each interior angle of a square and a rectangle is $90^{\circ}$
• Since both squares and rectangles are quadrilaterals, the sum of all interior angles of a square and a rectangle is $360^{\circ}$
• The diagonal of a square and a rectangle divides them into two right-angled triangles
• As both square and rectangle are parallelograms, the opposite sides of a square and a rectangle are parallel

### Difference Between Square and Rectangle

The following are the difference between a square and a rectangle

## The Length of the Diagonals in a Square is Equal

In a square, the length of the diagonals in a square is equal.

In the above figure, $\text{ABCD}$ is a square and $\text{AC}$, $\text{BD}$ are the diagonals.

Let’s understand the proof of the statement – the length of the diagonals in a square is equal, i.e., $\text{AC} = \text{BD}$.

Consider $\triangle \text{ABC}$ and $\triangle \text{BCD}$

$\text{AB} = \text{CD}$ (Sides of a square)

$\text{BC} = \text{BC}$ (Common)

$\angle \text{ABC} = \angle \text{BCD}$ (Each $90^{\circ}$)

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

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

## Each Diagonal Divide the Square into Two Congruent Triangles

Consider a square $\text{ABCD}$, where $\text{AC}$ and $\text{BD}$ are the two diagonals of the square.

• Diagonal $\text{AC}$ divides the rectangle $\text{ABCD}$ into two triangles – $\triangle \text{ABC}$ and $\triangle \text{ACD}$ and $\triangle \text{ABC} \cong \triangle \text{ACD}$
• Diagonal $\text{BD}$ divides the rectangle $\text{ABCD}$ into two triangles – $\triangle \text{ABD}$ and $\triangle \text{BCD}$ and $\triangle \text{ABD} \cong \triangle \text{BCD}$

Let’s prove that $\triangle \text{ABC} \cong \triangle \text{ACD}$

In $\triangle \text{ABC}$ and $\triangle \text{ACD}$

$\text{AB} = \text{CD}$ (Sides of a square)

$\text{BC} = \text{DA}$ (Sides of a square)

$\text{AC} = \text{AC}$ (Common)

Therefore, $\triangle \text{ABC} \cong \triangle \text{ACD}$ (SSS Congruence Criterion)

Let’s now prove that $\triangle \text{ABD} \cong \triangle \text{BCD}$

In $\triangle \text{ABD}$ and $\triangle \text{BCD}$

$\text{AB} = \text{CD}$ (Sides of a square)

$\text{AD} = \text{BC}$ (Sides of a square)

$\text{BD} = \text{BD}$ (Common)

Therefore, $\triangle \text{ABD} \cong \triangle \text{BCD}$ (SSS Congruence Criterion)

## The Diagonals of a Square Bisect Each Other at $90^{\circ}$

Consider a square $\text{ABCD}$, where $\text{AC}$ and $\text{BD}$ are the two diagonals intersecting at $\text{O}$, then

• $\text{OA} = \text{OC}$ and $\text{OB} = \text{OD}$
• $\angle \text{AOD} = \angle \text{BOC} = 90^{\circ}$

Let’s prove $\text{OA} = \text{OC}$ and $\text{OB} = \text{OD}$

In $\triangle \text{AOD}$ and $\triangle \text{BOC}$

$\text{AD} = \text{BC}$ (Sides of a square)

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

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

Therefore, $\triangle \text{AOD} \cong \triangle \text{BOC}$ (ASA Congruence Criterion)

Thus, $\text{OA} = \text{OC}$ and $\text{OB} = \text{OD}$ (Corresponding Parts of Congruent Triangles)

Now, let’s prove $\angle \text{AOD} = \angle \text{BOC} = 90^{\circ}$

Consider $\triangle \text{AOD}$ and $\triangle \text{COD}$

$\text{AD} = \text{CD}$ (Sides of a square)

$\text{OA} = \text{OC}$ (Diagonal of a square is bisected at the point of intersection)

$\text{OD} = \text{OD}$ (Common)

Therefore, $\triangle \text{AOD} \cong \triangle \text{COD}$ (SSS Congruence Criterion)

Thus, $\angle \text{AOD} = \angle \text{COD}$ ————————— (1)

Also, $\angle \text{AOD} + \angle \text{COD} = 180^{\circ}$ ($\text{AOC}$ is a straight line) —————— (2)

From (1) and (2) $\angle \text{AOD} + \angle \text{AOD} = 180^{\circ} => 2 \angle \text{AOD} = 180^{\circ} => \angle \text{AOD} = 90^{\circ}$

Therefore, $\angle \text{AOD} = \angle \text{COD} = 90^{\circ}$

## Practice Problems

1. What is a square in geometry?
2. State True or False
• Opposite sides of a square are equal
• Adjacent sides of a square are equal
• Opposite sides of a square are parallel
• Adjacent sides of a square are parallel
• Opposite sides of a square are perpendicular
• Adjacent sides of a square are perpendicular
• Opposite angles of a square are equal
• Adjacent angles of a square are equal
• Opposite angles of a square are complementary
• Adjacent angles of a square are complementary
• Opposite angles of a square are supplementary
• Adjacent angles of a square are supplementary
• The diagonals of a square are parallel to each other
• The diagonals of a square are perpendicular to each other
• The diagonals of a square are equal
• The diagonals of a square are unequal

## FAQs

### What is a square in geometry?

A quadrilateral that is made of four equal sides, with all the interior angles measuring $90^{\circ}$, is a square.

### Is the side of a square and it’s diagonal the same length?

No, the side of a square and its diagonal aren’t of the same length. The diagonal of a square is greater in length than its side.

### What are the 5 properties of a square?

The five important properties of a square are
a) All four sides of the square are equal to each other
b) The interior angles of a square measure $90^{\circ}$
c) Each diagonal divide the square into two congruent triangles
d) The length of the diagonals in a square is equal
e) The diagonals of a square bisect each other at $90^{\circ}$

### What are the properties of the diagonal of a square?

The important properties of the diagonals in a square are
a) Each diagonal divide the square into two congruent triangles
b) The length of the diagonals in a square is equal
c) The diagonals of a square bisect each other at $90^{\circ}$

## Conclusion

A square is a closed 2D shape with all four sides equal. Each of the four angles of the square measures $90^{\circ}$. There are two equal diagonals in a square that bisect each other at right angles.