# Squares

Squares are one of the most common shapes that we can find, in fact it is one of the first shapes that we learn at a very young age. There are many things around us that can have a square shape, for example, windows, stamps, chocolate, one side of a die, a bread slice, an alarm clock, a pizza box, a cushion, a handbag, among many others.

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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.

## 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:

Examples of objects with square shapes - Pixabay

## 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 $\overline{)DA}\parallel \overline{)BC}$.

Properties of a square (parallel opposite sides) - StudySmarter Originals

• 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 $\left(\overline{)AB}\cong \overline{)BC}\cong \overline{)CD}\cong \overline{)DA}\right)$.
Properties of a square (congruent sides) - StudySmarter Originals
• 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°.

Properties of a square (right angles) - StudySmarter Originals

• 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 $\overline{)AM}\cong \overline{)MC}\cong \overline{)BM}\cong \overline{)MD}$.

Properties of a square (diagonals) - StudySmarter Originals

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.

Length of the diagonal of a square - StudySmarter Originals

Using the Pythagoras theorem, we get the following:

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

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 $s\sqrt{2}$.

s2

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

Properties of a square (congruent isosceles triangles) - StudySmarter Originals

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 $△ABC$, and the blue one $△ADC$. 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.

Diagram representing relationships between quadrilaterals - StudySmarter Originals

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:

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

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

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

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

5. 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:

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

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:

$P=4·a\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=4·4cm\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\overline{)\mathbf{P}\mathbf{=}\mathbf{16}\mathbf{}\mathbf{c}\mathbf{m}}$

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.

$P=4·a=20m$

Now, we can solve for a:

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

### 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 cm2, m2, 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

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

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:

$P=4·a=24m$

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.

#### Flashcards in Squares 12

###### Learn with 12 Squares flashcards in the free StudySmarter app

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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 a2.

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:

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

## Test your knowledge with multiple choice flashcards

A square is a special case of a parallelogram. Is this true or false?

A square has all the properties of a rectangle, but not the ones from a rhombus. Is this true or false?

A square, a rectangle and a rhombus are all special cases of a quadrilateral, except for the parallelogram. Is this true or false?

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