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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Test yourself on square for 9th grade!

Look at the square below:

Is a parallelogram a square?

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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

Look at the square below:

Is a square a parallelogram?

Question 2

Look at the square below:

Is a rhombus a square?

Question 3

Look at the square below:

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
$A=a^2$

Square

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

Look at the square below:

Is a square a rectangle?

Question 2

Look at the square below:

Is a trapezoid a square?

Question 3

Is a square a trapezoid?

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

Square

Check your understanding

Question 1

Look at the square below:

Is a deltoid a square?

Question 2

Look at the square above:

Is a square a deltoid?

Question 3

Look at the square below:

What is the area of the square equivalent to?

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

Look at the square below:

What is the area of the square?

Question 2

Given the square:

What is the area of the square?

Question 3

Look at the square below:

What is the area of the 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

Look at the square below:

What is the area of the square?

Question 2

Look at the square below:

What is the area of the square?

Question 3

Look at the square below:

Is a parallelogram a square?

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

Look at the square below:

Is a square a parallelogram?

Question 2

Look at the square below:

Is a rhombus a square?

Question 3

Look at the square below:

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:

Square

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

Exercise #1

Look at the square below:

Is a parallelogram a square?

Step-by-Step Solution

To solve this problem, we need to understand the definitions and properties of a parallelogram and a square:

A parallelogram is a quadrilateral where opposite sides are parallel. This implies that opposite sides are equal in length, but it does not require all sides to be equal or all angles to be right angles.
A square is a special type of parallelogram and rectangle where all four sides are of equal length, and all four angles are right angles (90 degrees).

With these definitions in mind, let's compare:

A parallelogram, by definition, does not require all sides to be equal or all angles to be right angles. Therefore, not every parallelogram meets the requirements to be a square.

For example, a rectangle is a type of parallelogram where all angles are right angles, but it may not have all equal sides unless it is a square. Similarly, a rhombus is a type of parallelogram with all sides equal but may not have all right angles unless it is a square.

Thus, while a square is indeed a parallelogram (since it fulfills the conditions of having opposite sides equal and parallel), not every parallelogram is a square. Only those parallelograms which have all sides equal and all angles equal to 90 degrees qualify as squares.

This leads us to conclude that the statement "A parallelogram is a square" is false.

Therefore, the correct answer is No.

Answer

Exercise #2

Look at the square below:

Is a square a parallelogram?

Step-by-Step Solution

To determine if a square is a parallelogram, we must first define both geometric shapes.

Square: A square is a quadrilateral with four equal sides and four right angles. This means that all angles are $90^\circ$ and each pair of opposite sides are parallel.
Parallelogram: A parallelogram is a quadrilateral where opposite sides are parallel and equal in length. It does not necessarily require right angles.

Now, let's see if a square fits the definition of a parallelogram:

The square has opposite sides that are both parallel and equal, satisfying the definition of a parallelogram.
Although a square also has additional properties, such as all angles being right angles and all sides being equal, these characteristics do not contradict the definition of a parallelogram.

Since a square satisfies all the conditions required for a parallelogram, we conclude that a square is indeed a type of parallelogram.

Therefore, the answer to the problem is Yes.

Answer

Yes

Exercise #3

Look at the square below:

Is a rhombus a square?

Step-by-Step Solution

To determine whether a rhombus is a square, we must understand the properties of each shape.

Definition of a Rhombus:
A rhombus is a quadrilateral with all four sides of equal length. It may have angles that are not right angles.

Definition of a Square:
A square is a quadrilateral with all four sides of equal length and all four angles equal to $90^{\circ}$ .

Comparison:

Both a square and a rhombus have four equal sides.
A square has $90^{\circ}$ angles, whereas a rhombus does not necessarily have $90^{\circ}$ angles.

Thus, while all squares are rhombuses (because they have equal side lengths), not all rhombuses are squares (because not all rhombuses have right angles).

Therefore, a rhombus is not a square as a general statement.

Answer

Exercise #4

Look at the square below:

Is the square a rhombus?

Step-by-Step Solution

To solve this problem, we'll consider the definitions:

A square is a quadrilateral with four equal sides and four equal angles (each angle is $90$ degrees).
A rhombus is a quadrilateral with all sides equal in length.

Notice that for a quadrilateral to be a rhombus, it simply requires all sides to be equal, without any condition on the angles. Since a square has all four sides equal, it meets the fundamental requirement of a rhombus.

Therefore, every square can be classified as a rhombus because it satisfies the condition that all sides are equal.

Hence, the correct answer is Yes, the square is a rhombus.

Answer

Yes

Exercise #5

Look at the square below:

Is a square a rectangle?

Step-by-Step Solution

In this problem, we need to determine if a square meets the criteria for being classified as a rectangle. We start by examining the definitions:

Definition of a Rectangle: A rectangle is a quadrilateral with four right angles. Additionally, opposite sides are equal in length.
Definition of a Square: A square is a quadrilateral where all four sides are equal in length, and all four angles are right angles.

By examining these properties, we can see the following:

Since a square has four right angles, it satisfies the angle condition of a rectangle.
A square has all sides equal, which means opposite sides are also equal. This satisfies the side condition of a rectangle.

Therefore, since a square fulfills both the angle and opposite sides conditions required by the definition of a rectangle, a square is indeed a rectangle.

The correct answer to the question is: Yes.

Answer

Yes

Check your understanding

Question 1

Look at the square below:

Is a square a rectangle?

Question 2

Look at the square below:

Is a trapezoid a square?

Question 3

Is a square a trapezoid?

Start practice

Square

What is a square?

Square

Test yourself on square for 9th grade!

Properties of the Square

Proof of the Square

Area of the Square

Square

What is a square?

Let's look at all the properties in an illustration

Square

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

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

First step

Second step

Third step

How do you prove that a rhombus is a square?

How do you prove that a rectangle is a square?

From Parallelogram to Square

Area of the Square

Square

Square: Examples and Exercises with Solutions

Exercise #1

Step-by-Step Solution

Answer

Exercise #2

Step-by-Step Solution

Answer

Exercise #3

Step-by-Step Solution

Answer

Exercise #4

Step-by-Step Solution

Answer

Exercise #5

Step-by-Step Solution

Answer

More Questions

Area of the Square

Properties, Characteristics and proofs