Is a square a rectangle

Is a square a rectangle
Go back to  'Quadrilaterals'

There's a park in front of Raman's house. Raman assumed it was rectangular in shape. At last, that's what he thought! 

One fine day, while playing, he noticed something interesting. It took him the same amount of time to cycle around the length of the park as it took him to cycle across the width of the park. This raised a doubt in his mind. Is the park a square, or a rectangle, or more surprisingly, both? 

Square park

You will find out the answer on this page.

You will also find out answers to the questions such as, is a rhombus a rectangle, is a square a parallelogram and is square a trapezoid.

Check out the interactive simulation to know more about the lesson and try your hand at solving a few interesting questions at the end of the page.

Find out the answer to "Is a square a rectangle" in just 6 minutes!

Lesson Plan


What Are Squares and Rectangles?

Let us first discuss a rectangle.

A rectangle is a two-dimensional, four-sided plane figure with four interior right angles and four corners. The opposites sides of a rectangle are equal.

Rectangle

A square is a two-dimensional plane figure with four equal sides, four interior right angles, and four corners.

A Square

Now, we will discuss the properties of squares and rectangles in detail.


What are the Properties of Squares and Rectangles?

Square Rectangle
All four interior angles are \(90^{\circ}\) All four interior angles are \(90^{\circ}\)    
All four sides of the square are equal to each other. The opposite sides of a rectangle are equal.
Opposite sides of a square are parallel to each other. Opposite sides of a rectangle are parallel to each other.    

The diagonals of a square bisect each other at \(90^{\circ}\).

Also, the diagonals of a square are equal.

The diagonals of a rectangle bisect each other.

Also, the diagonals of a rectangle are equal.

The perimeter of a square is \(4 \times \text{ side}\). The perimeter of a rectangle is \(2(\text{length+breadth})\).
Area of a square is \(\text{side}^{2}\). Area of a rectangle is \(\text{length }\times \text{breadth}\).

Properties of quadrilaterals: Properties of a square

Properties of quadrilaterals: Properties of a rectangle

 


What are Quadrilaterals?

Let us now study the concept of quadrilaterals.

A quadrilateral can be defined as a closed, two-dimensional figure that is formed by joining four points among which any three points are non-collinear.

For example, the following figures are quadrilaterals.

What is a quadrilateral? Quadrilateral Example

What is a quadrilateral? Quadrilateral Example

We can observe that a quadrilateral has four sides, four angles, and four vertices.

When we have to name a quadrilateral, we have to keep in mind the order of the vertices.

For example, consider the following quadrilateral:

What is a quadrilateral? Naming conventions for quadrilaterals

We could name this quadrilateral \(ABCD, BCDA, ADCB, DCBA\), etc., but we cannot name it \(ACBD, BACD\), etc. since, in the latter names, the order of the vertices is incorrect.

A quadrilateral \(ABCD\) will have two diagonals: \(AC\) and \(BD\).

Properties of Quadrilaterals

  • They have four sides.
  • They have four vertices.
  • They have two diagonals.
  • The sum of all interior angles is \(360^{\circ}\).

What are the Different Types of Quadrilaterals?

We can understand the hierarchy of quadrilaterals using the following chart:

Properties of quadrilaterals: Quadrilaterals chart hierarchy

The above figure shows the family tree of different types of quadrilaterals.

As indicated in the above figure, a square comes under a rectangle, that is, a square is a rectangle with all equal sides.

Let us go through different types of quadrilaterals.

Quadrilateral Definition Shape (With Highlighted Properties )
Square

A square is a quadrilateral with four equal sides and four right angles.

Properties of quadrilaterals: Properties of a square
Rectangle

A rectangle is a quadrilateral with two pairs of equal and parallel opposite sides and four right angles.

Properties of quadrilaterals: Properties of a rectangle
Parallelogram

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. 

Properties of quadrilaterals: Properties of a parallelogram
Rhombus

A rhombus is a quadrilateral with four equal sides.

Properties of quadrilaterals: Properties of a rhombus
Trapezium

A trapezium is a quadrilateral in which one pair of opposite sides is parallel. 

Properties of quadrilaterals: Properties of a trapezium
Kite

A kite is a quadrilateral in which two pairs of adjacent sides are equal. 

Properties of quadrilaterals: Properties of a kite

 

 
important notes to remember
Important Notes
  1. All squares are rectangles, but not all rectangles are squares.
  2. All squares are rhombuses, but not all rhombuses are squares.

Why a Square is a Special Type of Rectangle?

A square is a special type of rectangle because it possesses all the properties of a rectangle.

Similar to a rectangle, a square has:

  • interior angles which measure \(90^{\circ}\) each.
  • opposite sides that are parallel and equal.
  • two diagonals that bisect each other and are equal.

A square is called a special kind of rectangle because it possesses some additional properties which do not apply to rectangles.

  • All four sides of a square are equal.
  • Diagonals of a square bisect each at right angles. 

rectangle, square, is a square a rectangle

 
Thinking out of the box
Think Tank
  1. Is every four-sided figure a quadrilateral?
  2. Are all rectangles squares?
  3. Are all rhombuses squares?
  4. Is a square a trapezoid?
  5. Is a square a parallelogram?

Solved Examples

Let us now look at some solved examples.

Example 1

 

 

A teacher placed a few food items in front of her students.

She asked them to identify the food items that were similar in shape to a quadrilateral.

Can you identify them?

Shapes around us - A half-sandwich, a donut, a cracker, a fried egg

Solution

A half-sandwich, a doughnut, a cracker and an egg are shown in the above image.

We have to identify the items that are shaped similar to a quadrilateral.

A quadrilateral has four sides. 

Hence, the cracker is the only food item that has the shape of a quadrilateral.

\(\therefore\) Cracker is the only quadrilateral-shaped item.
Example 2

 

 

There is a square park near Sneha's house.

The length of each side of the park is \(300 \text{ metres }\).

Find the perimeter of the park.

Square park

Solution

The length of each side of the park is \(300 \text{ metres}\).

The perimeter of a square is \(4 \times \text{side}\).

Therefore, the perimeter of the park is:

\[4 \times 300=1200 \text{ metres}\]

\(\therefore\) Perimeter is \(1200 \text{ metres}\).
Example 3

 

 

Find the area of the floor of a square room which is made up of \(100\) tiles of side \( 15 \text{ centimetres}\) each.

Examples for the area of a square formula - Find the area of the square floor.

Solution

Area of one tile is:

\[15 \text{ cm } \times 15 \text{ cm } = 225 \text{ cm }^2\]

We know that there are \(100\) tiles on the floor of the room.

Thus, the area occupied by \(100\) tiles is:

\[100 \times 225 \text{ cm }^2= 22500 \text{ cm }^2\]

We know,

\( 10000 \text {cm} ^2 =  1 \text m^2\)

Thus,

22500 \( \text{cm}^2 = \frac{22500}{10000} = 2.25 \text m^2 \)

\(\therefore\)  Area is \( 22500 \text{ cm }^2 = 2.25 \text m^2 \)
Example 4

 

 

The length and breadth of a rectangular carpet is \(15 \text{ metres }\)and \(10 \text{ metres}\).

Find the area covered by the carpet.

Area of a rectangle, Area of a rectangular carpet

Solution

Area of a rectangle is \(\text{ length}\times \text{ breadth }\).

Area of the carpet is:

\[15 \times 10= 150 \text{ square metres}\]

\(\therefore\)  Area is \(150 \ \text {square metres }\)
Example 5

 

 

Quadrilateral shapes

A teacher asked the students in her class to identify the category of the shapes given above.

Can you find the right answer?

Solution

Every shape in the given image is a closed figure and has four sides.

Therefore, they can all be categorized as quadrilaterals.

\(\therefore\)  The shapes are quadrilaterals.

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Let's Summarize

We hope you enjoyed finding out the answer to the question - 'Is a Square a Rectangle' with the simulation and practice questions. Now you will be able to easily solve problems and answer questions, such as, is a rhombus a rectangle, is a square a parallelogram, and is square a trapezoid.

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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs)

1. A square is a rectangle. Is this statement true or false?

This statement is true. A square is a rectangle because it possesses all the properties of a rectangle.

2. What is the difference between a square and a rectangle?

Squares possess some additional properties which do not apply to rectangles. 

A square has all four sides equal whereas, in a rectangle, the opposite sides are equal.

3. Is a square a rectangle and a rhombus?

A square is both a rectangle and a rhombus because it possesses the properties of rectangles and rhombuses. If you are also looking for the answer to the question - is a rhombus a rectangle, the answer is no. 

  
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