Square

What is a square?

A quadrilateral whose sides (or edges) are all equal and all its angles are also equal, is a square.
Furthermore, a square is a combination of a parallelogram, a rhombus, and a rectangle.
Therefore, the square has all the properties of the parallelogram, the rhombus, and the rectangle.

Detailed explanation

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Given the square, is it a trapezoid?

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Properties of the Square

All sides of the square are equal.
All angles of the square measure $90^o$ degrees.
In a square, there are two pairs of opposite sides that are parallel.
The diagonals of the square intersect, are perpendicular, and equal.
The diagonals of the square are bisectors.

Proof of the Square

If all sides and angles of a quadrilateral are equal, we can determine that it is a square.
How can we prove that a quadrilateral is a square if we have no data?
We will proceed in the following order:

We will prove that the quadrilateral is a parallelogram.
We will prove that the parallelogram is a rectangle or rhombus.
We will prove that the rectangle or rhombus is a square.

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Test your knowledge

Question 1

Given the square:

Is the parallelogram a square?

Question 2

Given the square:

Is a rhombus a square?

Question 3

Given the square:

Is the square a rhombus?

Area of the Square

The area of a square is equal to the length of one of its sides multiplied by itself.

We will indicate the area of the square with the letter $A$
and the side (or edge) of the square with the letter $a$ .
The formula will be:
$A=a\times a$
or
$A=a^2$

The square is a very special figure.
In this article, we will learn everything that needs to be known about the square and discover its incredible properties.

Do you know what the answer is?

Question 1

Given the square:

Is a deltoid a square?

Question 2

Given the square:

Is a square a deltoid?

Question 3

Given a square:

Is a square a rectangle?

What is a square?

The square is a combination of a parallelogram, a rhombus, and a rectangle
Therefore, the square has all the properties of the parallelogram, the rhombus, and the rectangle.
We will summarize all its properties here:

All sides of the square are equal.
All angles of the square are 90 degrees.
In a square, there are two pairs of opposite sides that are parallel.
The diagonals of the square intersect, are perpendicular, and equal.
The diagonals of the square are bisectors.

Let's look at all the properties in an illustration

Check your understanding

Question 1

Given the square:

Is the square a parallelogram?

Question 2

Given the square:

Is a trapezoid a square?

Question 3

The quadrilateral ABCD is a square.

Is the square a rectangle?

How can you prove that a certain quadrilateral is a square?

If you have a quadrilateral in front of you and you know that all its sides (or edges) are equal and all its angles are also equal, you can conclude that it is a square!

Moreover, in certain cases, you can prove that the quadrilateral you have is a rhombus or a rectangle and, from this point, you can find the property that proves that said rectangle or rhombus is a square.

How can we prove that a quadrilateral is a square if we don't have any data?

First step

We will demonstrate that the quadrilateral is a parallelogram.

Don't remember how to prove that a certain quadrilateral is a parallelogram?
Reminder of proofs:

Any quadrilateral whose opposite sides are also parallel is a parallelogram.
Any quadrilateral whose opposite sides are also equal to each other, is a parallelogram.
If in a certain quadrilateral there is a pair of opposite sides that are equal and also parallel, the quadrilateral is a parallelogram.
If the diagonals of the quadrilateral intersect, it is a parallelogram.
If in a certain quadrilateral there are two pairs of opposite angles that are equal, it is a parallelogram.

Do you think you will be able to solve it?

Question 1

Given the quadrilateral ABCD:

Is the aforementioned quadrilateral necessarily a square?

Question 2

Given the quadrilateral ABCD:

Is the aforementioned quadrilateral necessarily a square?

Question 3

Given the quadrilateral ABCD:

Is the quadrilateral above necessarily a square?

Second step

We will demonstrate that the parallelogram is a rectangle or rhombus.

Third step

We will demonstrate that the rectangle or rhombus is a square.

Test your knowledge

Question 1

Given the quadrilateral ABCD:

Is the quadrilateral above necessarily a square?

Question 2

Given the quadrilateral ABCD:

Is the quadrilateral above necessarily a square?

Question 3

Given the square, is it a trapezoid?

How do you prove that a rhombus is a square?

If you have a rhombus in front of you, you can determine that it is a square if at least one of the following conditions is met:

If the rhombus has a right angle (90 degrees), you can conclude it's a square!
If the diagonals of the rhombus are equal you can conclude it's a square!

How do you prove that a rectangle is a square?

If you have a rectangle in front of you, you can determine that it is a square if at least one of the following conditions is met:

If the rectangle has a pair of equal adjacent sides you can conclude it's a square!
If the diagonals of the rectangle are perpendicular you can conclude it's a square!
If the diagonals of the rectangle are bisectors you can conclude it's a square!

Useful information:
Is every square a rectangle?
Yes!

Is every rectangle a square?
Definitely not.

Is every square a rhombus?
Yes!

Is every rhombus a square?
No, no, and no.

Do you know what the answer is?

Question 1

Given the square:

Is the parallelogram a square?

Question 2

Given the square:

Is a rhombus a square?

Question 3

Given the square:

Is the square a rhombus?

From Parallelogram to Square

How can you prove that a certain parallelogram is a square?
To start, we must prove that the parallelogram is a rhombus or a rectangle.
Then, based on the conditions specified above, we must prove that the rhombus or rectangle is a square.

Area of the Square

Calculating the area of a square is very simple and is similar to calculating the area of a rectangle.
To calculate the area of a square we will multiply side by side.
In a square, all sides are equal and, therefore, the area of the square will be equal to side squared or side by side (which in fact is one side multiplied by itself).
Let's see it illustrated and in the formula:

We will indicate the area of the square with the letter $A$
The formula will be:
$A=a\times a$
or
$A=a^2$

Square: Examples and Exercises with Solutions

examples.example_title

Given the quadrilateral ABCD:

Is the mentioned quadrilateral necessarily a square?

examples.explanation_title

As we see that BD is equal to 8 and AC is equal to 7, the sides are not equal, and this contradicts the properties of the square, where all sides are equal to each other, therefore the quadrilateral is not a square

examples.solution_title

examples.example_title

Given the square:

Is AF necessarily equal to ED with the available data?

examples.explanation_title

Since it is not given that FD is parallel to AE, it cannot be argued that AF is necessarily equal to ED

examples.solution_title

examples.example_title

Given the square ABCD and the parallelogram AFED

The area of the parallelogram is equal to 100

ED=5

How long is AC?

examples.explanation_title

The area of a parallelogram is equal to the side multiplied by the height, as we know that ED is equal to 5, the area is also given.

We place the data in the following formula:

$S=AC\times ED$

$100=AC\times5$

We divide by 5 the two sections:

$\frac{100}{5}=\frac{5AC}{5}$

$20=AC$

examples.solution_title

$20$

examples.example_title

Given the square:

What types of triangles do the diagonals in the square form?

examples.explanation_title

The diagonals of the square intersect each other, so the four triangles are isosceles. Moreover, since the diagonals are perpendicular to each other, the diagonals form four right-angled triangles. Therefore, the correct answers are A+C

examples.solution_title

Answers a and c are correct

examples.example_title

Given the square:

Is BE equal to CE?

examples.explanation_title

According to the properties of the square, the diagonals intersect each other, therefore, BE is equal to CE

examples.solution_title

Yes

Check your understanding

Question 1

Given the square:

Is a deltoid a square?

Question 2

Given the square:

Is a square a deltoid?

Question 3

Given a square:

Is a square a rectangle?

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