In this article, we will define what a square is, its properties, the formulas for the area and perimeter of a square, and examples of their application.

## Definition of a Square

A **square** is a quadrilateral with all its sides and all its angles equal in measure.

Read about Quadrilaterals to refresh the basics.

## Examples of squares

As mentioned at the beginning of this article, we can find square shapes in many things around us. Let's show you a few examples.

A picture frame, a gift box, a chess board and a pizza box are all objects that include square shapes:

## Properties of a square

A square is a special case of a parallelogram, therefore it has the same properties that a parallelogram has, but it also has other unique properties that make it the shape it is. Read more about Parallelograms, if you need a recap.

We can define the **properties of a square** as follows:

Like a parallelogram, a square has

**all its opposite sides parallel to each other**. In the example below, $\overline{)AB}\parallel \overline{)CD}$ and $\overline{)DA}\parallel \overline{)BC}$.

**All the sides of a square are congruent**, which means that they measure the same. In the example below, all sides of the square measure 2 units, therefore they are all congruent $(\overline{)AB}\cong \overline{)BC}\cong \overline{)CD}\cong \overline{)DA})$.

**All four angles of a square are equal to 90°****(right angles)**. In the image below, $\angle A,\angle B,\angle Cand\angle D$ all measure 90°.

The

**diagonals of a square**are**equal in length and bisect each other at an angle of 90°**. In other words, the diagonals are perpendicular to each other, and intersect in their middle. We can notice in the figure below that the two diagonals $\overline{)AC}$ and $\overline{)BD}$ intersect at a point M, and are perpendicular to each other $\overline{)AC}\perp \overline{)BD}$. Since $\overline{)AC}\cong \overline{)BD}$, M is the middle of both diagonals, therefore $\overline{)AM}\cong \overline{)MC}\cong \overline{)BM}\cong \overline{)MD}$.

The **length of the diagonal of a square,** in relation to the length of its side, can be calculated using the Pythagoras theorem.

Remember that, the **Pythagoras theorem** states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two other sides in a right triangle.

Let's work out the length of the diagonal $\overline{)AC}$, which we will call **d**, given that the length of the side of the square equals **s**, as shown in the image below.

Using the Pythagoras theorem, we get the following:

${d}^{2}={s}^{2}+{s}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{d}^{2}=2{s}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\sqrt{{d}^{2}}=\sqrt{2{s}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{)\sqrt{d\overline{){}^{2}}}}=\sqrt{2}\xb7\overline{)\sqrt{{s}^{\overline{)2}}}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}d=\sqrt{2}\xb7s\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{)\mathbf{d}\mathbf{=}\mathbf{s}\sqrt{\mathbf{2}}}$

In a square, the length of a diagonal is equal to the length of the side multiplied by the square root of 2. Therefore, the length of both diagonals: $\overline{)AC}$ and $\overline{)BD}$ is equal to $s\sqrt{2}$.

A

**diagonal of a square**divides it into**two congruent isosceles triangles**. This property can be illustrated in the image below:

In the square illustrated in the image above, we can notice that the diagonal $\overline{)AC}$ divided our square into two triangles: the green one $\u25b3ABC$, and the blue one $\u25b3ADC$. These two triangles are isosceles, since each triangle has two equal sides.

### Common properties between squares, rectangles, parallelograms and rhombuses

In the diagram below, you can see that a square is a special case of a rectangle and a rhombus.

In other words, **all squares are rectangles and rhombuses**. This is true since a square has all the properties of both a rectangle and a rhombus, along with some extra properties. A **summary of the properties of quadrilaterals** can be seen in the table below:

Properties of a quadrilateral | Rectangle | Square | Parallelogram | Rhombus |

All sides are equal | ||||

Opposite sides are equal | ||||

Opposite sides are parallel | ||||

All angles are equal | ||||

Opposite angles are equal | ||||

Sum of two adjacent angles is 180° | ||||

Diagonals bisect each other | ||||

Diagonals bisect each other perpendicularly |

We can draw the following **conclusions from the diagram and the table above**:

A square has all the properties of a rectangle and a rhombus;

A square is a special case of a rectangle, but also a special case of a rhombus;

A square, a rectangle and a rhombus have all the properties of a parallelogram;

A square, a rectangle and a rhombus are all special cases of parallelograms;

A square, a rectangle, a rhombus and a parallelogram are all special cases of quadrilaterals.

## Square formulas

The formulas related to squares that you need to remember are the ones to calculate their **perimeter** and **area**. Let's see some examples of how to calculate both in the following sections.

### Perimeter of a square

A two-dimensional item is any shape that can be put on a flat surface. The **perimeter** of any two-dimensional shape is **the length of its boundary** or sides.

The square is a 2D shape with four equal sides and four 90**°** angles. Therefore, we can define its perimeter as follows.

The **perimeter of a square** is the sum of the lengths of its four sides.

In other words, if a square has 4 sides of length **a**, then its **perimeter (P)** can be calculated as follows:

$P=a+a+a+a\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}P=4\xb7a$

Let's explore this in more detail with some examples.

**If a square has its sides equal to 4 cm each. What will be its perimeter?**

**Solution**

In this case, the length's side of the square **a** is equal to 4 cm. Its perimeter, or the sum of the lengths of all its sides, will be equal to:

$P=4\xb7a\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=4\xb74cm\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{)\mathbf{P}\mathbf{=}\mathbf{16}\mathbf{}\mathbf{c}\mathbf{m}}$

**If the perimeter of a square is 20 m, then what is the length of the side of the square?**

**Solution**

We need to find in this case the length of the side of a square: **a**.

We know that the perimeter of the square is 20 m, and the formula of perimeter is 4 times the side's length **a**.

$P=4\xb7a=20m$

Now, we can **solve for a**:

$\frac{\overline{)4}\xb7a}{\overline{)4}}=\frac{20m}{4}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{)\mathbf{a}\mathbf{=}\mathbf{}\mathbf{5}\mathbf{}\mathbf{m}}$

### Area of a square

In general, the **area** is defined as the **region contained inside the boundaries** of a flat object or 2D figure. In other words, the area is the amount of space occupied by the object. Specifically, the **area of a square** can be defined as follows:

The **area of a square** refers to the space contained within the boundaries of its four sides.

This measurement is given in square units, such as cm^{2}, m^{2}, etc.

The **formula to calculate the area of a square (A) **is:

$A={a}^{2}$,

where **a** is the length of the square's side.

**Determine the area of a square with a side length of 1.20 m.**

**Solution**

$A={(1.20m)}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{)\mathbf{A}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{44}\mathbf{}{\mathbf{m}}^{\mathbf{2}}}$^{ }

**Find the area of a square if its perimeter is equal to 24 m. **

**Solution**

First, we need to find the length of the side of a square **a**, to be able to calculate the area.

We know that the perimeter of the square is 24 m, and the formula of perimeter is 4 times the side's length **a**, so we get the following:

$P=4\xb7a=24m$

Again, we **solve for a**:

$\frac{\overline{)4}.a}{\overline{)4}}=\frac{24m}{4}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{)\mathbf{a}\mathbf{=}\mathbf{6}\mathbf{}\mathbf{m}}$

Then, the **area** will be equal to:

$A={a}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}={\left(6m\right)}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{)\mathbf{A}\mathbf{=}\mathbf{36}\mathbf{}{\mathbf{m}}^{\mathbf{2}}}$

## Squares - Key takeaways

A square is a quadrilateral with all its sides and all its angles equal in measure.

A square, a rectangle and a rhombus are special cases of a parallelogram.

A square, a rectangle, a rhombus and a parallelogram are all special cases of a quadrilateral.

- All four angles of a square are equal to 90°.
- All four sides of the square are congruent or equal to each other.
- The opposite sides of the square are parallel to each other.
- The diagonals of the square bisect each other at 90°.
- The perimeter of a square is equal to its side's length multiplied by four.
- The area of a square is equal to its side's length, raised to the power of two.

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##### Frequently Asked Questions about Squares

What is a square?

A square is a quadrilateral with all its sides and all its angles equal in measure.

Is a square a rectangle?

A square is a special case of a rectangle, where all its sides are equal and its diagonal perpendicularly bisect each other.

What is the area of a square?

The area of a square is equal to its side's length raised to the power of 2. If the length of the square's side is equal to a, then its area is equal to a^{2}.

What is the perimeter of a 4 cm square?

If a square has its side equal to 4cm, then its perimeter will be equal to 4 times 4 cm, which equals 16 cm.

What are the 5 properties of a square?

The 5 properties of a square are:

- A square has all its opposite sides parallel to each other.
- All four sides of a square are congruent or equal to each other.
- All four angles of a square are equal to 90° (right angles).
- The diagonals of a square are equal in length and bisect each other at 90°.
- A diagonal of a square divides it into two congruent isosceles triangles.

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