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How to Find the Area of a Rectangle?

Area of rectangle

The area of rectangle is amongst one of the most important and popular parameters of a rectangle. In this article, we will cover major theories and mathematical formulas related to the area of a rectangle, like what is the area of a rectangle, the formula of the area rectangle, and its application in daily life.

Do we all know what a rectangle is? First, let us understand some fundamental concepts related to a rectangle. A rectangle is a closed figure made up of 4 sides, just like a square. But, unlike a square, a rectangle doesn’t have all sides equal. A rectangle is a four-sided figure where the opposite sides are equal. To understand this see the figure below:

You can see that the side marked as A is opposite and equal to the side marked as C. Similarly, the side marked as B is equal and opposite to the side marked as D. Hence, ABCD is a rectangle. 

“You can also note that a square is a special case of rectangles with all opposite sides equal.”

We can find rectangles everywhere in our daily life. The notebooks you write on to books you study are rectangles. Some excellent examples of rectangular shapes are the tiles installed in your home, the blackboard teacher uses in the school, the dining table, the shape of your television screen, and much more.  

Let us learn some of the mathematical terms and concepts related to a rectangle now:

  • A rectangle has only opposite sides as equal. Two adjacent sides are never equal in the case of a rectangle.
  • A rectangle is a parallelogram. This means that the opposite sides of a rectangle are parallel to each other or opposite sides never cross each other.
  • The diagonal of a rectangle divides it into two triangles of equal area. 
  • The perimeter of a rectangle: The distance covered by the boundary of a rectangle is known as the perimeter of a rectangle. It is mathematically formulated as 

P (rectangle) = 2 ( length + breadth)

Now that we are well versed with the basics of a rectangle let us find out the area of a rectangle and the formula of the area rectangle.

What is the Area of a Rectangle?

Before diving deep into the area of the rectangle, are we all acquainted with what ‘area’ means? The area is defined as the space swept or covered by any closed figure. The space present inside the boundary of any figure is termed the area of that figure. From this, we can say that the space swept by or covered by a rectangle is known as the area of the rectangle. In other words, the two-dimensional space inside the perimeter of a rectangle is its area. See the figure given below. The red line indicates the boundary of a rectangle and the yellow covered filled space is the area of the rectangle. 

Later in this article, we will see the formula of the area of rectangle formula. Using that formula, you can even determine the area of the floor of your house, the area of your computer or mobile screen, etc.

How do you find the Area of a Rectangle?

The number of unit squares which can perfectly fit inside a rectangle gives the area of that rectangle. You might have got confused with this definition, right? Don’t worry; let us clarify this for you.  

For instance, let us make a rectangle with length = 2 cm and breadth = 3 cm. Let us now try to fit squares of length 1 unit inside this rectangle.

“Unit length: 1 is known as the unit length. The unit may be cm, inches, m, feet, etc. but remember whenever unit length is written, always understand it as 1.”  

So squares of unit length mean that the length of each side of the square is one. As we can see in the figure below, 6 squares of unit length can easily fit inside this rectangle therefore, we can say that the area of the rectangle is 6 units. Also, we know that the sides of the rectangle are in cm; therefore, the area of the rectangle changes from 6 units to 6 cm. 

We have completed all the conceptual learning in the area of a rectangle. Next, let us study and derive the area of rectangle formula.

Area of a Rectangle Formula

The formula which we are going to learn now is one of the simplest formulas in mathematics. Despite being simple it is one of the most powerful formulas too. The area was found using this formula only from the table you are studying to the room you are dwelling in. 

The area of rectangular formula is given as:

Area = length x width or length x breath

The product of the length and breadth of a rectangle gives us the formula for the area of that rectangle. Let the length of the rectangle be ‘a’ and breath be ‘b’; therefore, area ‘A’ is written as 

A = a x b (square units)

Example: The length of a rectangular roof is 12 m, and the breadth is 7 m. Find the area of wood that would be used to cover the entire roof?

Solution: Given length ‘a’ = 12

                          breadth ‘b’ = 7 

               Area of wood required to cover the entire roof = Area of the roof  

                          A (roof) = a x b

                                       = 12 x 7

                                       = 84 m2

Now that we have learned what the formula of area of rectangle is, let us learn how to calculate it. 

Steps on how to Calculate Area of Rectangle?

The steps on how to calculate the area of a rectangle are mentioned below. If you follow these steps properly, you will never get errors in your solutions.

Step 1: Write down the dimensions of the given rectangle from the questions.

Step 2: Put the values in the area of rectangle formula, i.e., length x width.

Step 3: Multiply the values and get the product.

Step 4: Write the result in square units 

To further understand how to calculate the area of a rectangle, consider the following example. We’ll calculate the area of a rectangle with a length of 20 units and a width of 5 units.

Step 1: Given the length of the rectangle = 20 units, breadth of the rectangle = 5 units

Step 2: Area of rectangle formula = length x breadth

Step 3: 20 x 5 = 100

Step 4: The area of the rectangle is 100 square units. 

Proving Area of a Rectangle?

Throughout the entire article, we have been doing the area of a rectangle is ‘length × breadth’. But ever wondered why this is the formula? In this section, let us derive the area of a rectangle formula.

See the figure mentioned below. We can see a rectangle KLMN with a diagonal KM. This diagonal has divided the rectangle into two triangles of equal areas. Therefore the area of the rectangle = 2 x area of 1 triangle.  

Let us take the triangle KMN. The base KN of the triangle is the length of the rectangle, say ‘a’, and MN the breadth of the rectangle is the height of the triangle, say ‘b’. Since KMN is a right-angled triangle (because the adjacent sides of a rectangle are perpendicular to each other)

We know the area of a triangle is = ½ x base x height 

Therefore Area of KMN = ½ x a x b

Also, the area of rectangle = 2 x (area of KMN)

                                          = 2 x ½ x a x b

                                          = a x b 

Hence proved that the area of a rectangle is always its length x width.

Let us now suppose that we are not given the dimensions of the length of any rectangle; instead, we are given the length of any one of the diagonals and the breadth. How do we find out the area? We will find out in the next section of this article.

Area of a Rectangle by Diagonal

As mentioned earlier, the diagonal divides a rectangle into two right-angled triangles. We shall use the Pythagoras theorem to determine a formula to find the area of a rectangle using diagonals.

According to Pythagoras theorem :

(Hypotenuse)2 = (Base)2 + (Perpendicular height)2

In this case:

Let diagonal be ‘d’, length be ‘l’ and breadth be ‘b’.

(Diagonal)2 = (Length)2 + (Breadth)2

(d)2 = (l)2 + (b)2

(l)2 = (d)2 – (b)2

l = √(d)2 – (b)2

Substituting the value of l in the main formula 

Area of the rectangle = l x b 

                                   = {(d)2 – (b)2} x b 

 So the area of a rectangle when length of one diagonal and dimension of breadth are given is {(d)2 – (b)2} x b = {(Diagonal)2 – (Breadth)2} x Breadth 

Therefore, you can use either of the methods to find the area of the rectangle depending on the data you are given. 

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