Squares

Definition of a Square

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.

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 $(\overline{AB}\cong \overline{BC}\cong \overline{CD}\cong \overline{DA})$.

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.

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:

$$\begin{gathered} d^2=s^2+s^2 \\ {d}^2=2s^2 \\ \sqrt{d^2}=\sqrt{2s^2} \\ \cancel{\sqrt{d\cancel{{}^2}}}=\sqrt{2}·\cancel{\sqrt{s^{\cancel{2}}}} \\ d=\sqrt{2}·s \\ \boxed{\mathbf{d}\mathbf{=}\mathbf{s}\sqrt{\mathbf{2}}} \end{gathered}$$

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:

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.

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

$$\begin{gathered} P=a+a+a+a \\ P=4·a \end{gathered}$$

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

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:

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.