Rectangle Formula

Rectangle formulas include the formula for area, perimeter, and diagonal of a rectangle. To recall, a rectangle is a four sided polygon and the length of the opposite sides are equal. A rectangle is also called as an equiangular quadrilateral, as all the angles of a rectangle are right angled. A rectangle is a parallelogram with right angles in it. When the four sides of a rectangle are equal, then it is called a square.

Formula for Rectangle

There are mainly three formulas for rectangle – Perimeter, Area and Diagonal. The formulas for area, perimeter, and diagonal of a rectangle are:

Solved Questions Using Formulas of Rectangle

Question:

Find out the length of the rectangle if the area is 96 cm ² and the breadth is 16 cm.

Solution:

As we know, the Area of a rectangle = l × b

Here the area is already given, so find the length of the rectangle.

So,96= l × 16

=>l= 96 ⁄ 16 i.e. Length = 6 cm

What is the Area of Rectangle?

Definition: Area of rectangle is the region occupied by a rectangle within its four sides or boundaries.

The area of a rectangle depends on its sides. Basically, the formula for area is equal to the product of length and breadth of the rectangle. Whereas when we speak about the perimeter of a rectangle, it is equal to the sum of all its four sides. Hence, we can say, the region enclosed by the perimeter of the rectangle is its area. But in the case of a square, since all the sides are equal, therefore, the area of the square will be equal to the square of side-length.

Area of Rectangle Formula

The formula to find the area of rectangle depends on its length and width. The area of a rectangle is calculated in units by multiplying the width (or breadth) by the Length of a rectangle. Lateral and total surface areas can be calculated only for three-dimensional figures. We cannot calculate for the rectangle since it is a two-dimensional figure. Thus, the perimeter and the area of a rectangle is given by:

The area of any rectangle is calculated, once its length and width are known. By multiplying length and breadth, the rectangle’s area will obtain in a square-unit dimension. In the case of a square, the area will become side². The main difference between square and rectangle is that the length and breadth are equal for square.

How to Calculate the Area of a Rectangle

Follow the steps below to find the area:

Step 1:  Note the dimensions of length and width from the given data

Step 2: Multiply length and width values

Step 3: Write the answer in square units

Area of Rectangle Using Diagonal

We know that the diagonal of a rectangle is calculated using the formula:

(Diagonal)² = (Length)² + (Width)²

From this,

(Length)² = (Diagonal)² – (Width)²

Find a missing side length when the area is known

Formula: Length = Area/Width

Why Area of the rectangle is length x breadth?

The diagonals of the rectangle divide it into two equivalent right-angled triangles. Therefore, the area of the rectangle will be equal to the sum of the area of these two triangles.

Suppose, ABCD is a rectangle.

Now, let diagonal AC divide the rectangle into two right triangles, i.e. ∆ABC and ∆ADC.

We know that ∆ABC and ∆ADC are congruent triangles.

Area of ∆ABC = ½ x base x height =  ½ x AB x BC =  ½ x b x l

Area of ∆ADC = ½ x base x height = ½ x CD x AD =  ½ x b x l

Area of rectangle ABCD = Area of ∆ABC + Area of ∆ADC

Area (ABCD) = 2(½ x b x l)

Area (ABCD) = l x b

Thus, the area of the rectangle = Length x Breadth

Area of rectangle Solved Examples

Example 1: Find the area of the rectangle whose length is 15 cm and the width is 4 cm.

Solution:

Given,

Length = 15 cm

Width = 4 cm

Area of a rectangle = Length × Width

15 × 4  = 60

So the area of rectangle = 60 cm²

Example 2: Find the area of a rectangular blackboard whose length and breadth are 120 cm and 100 cm, respectively.

Solution:

Length of the blackboard = 120 cm = 1.2 m

Breadth of the blackboard = 100 cm = 1 m

Area of the blackboard = area of a rectangle = length x breadth = 1.2 m x 1 m = 1.2 square-metres

Example 3: The length of a rectangular screen is 15 cm. Its area is 180 sq. cm. Find its width.

Solution:

Area of the screen = 180 sq. cm.

Length of the screen = 15 cm

Area of a rectangle = length x width

So, width = area/length

Thus, width of the screen = 180/15 = 12 cm

Example 4: The length and breadth of a rectangular wall are 75 m and 32 m, respectively. Find the cost of painting the wall if the rate of painting is Rs 3 per sq. m.

Solution:

Length of the wall = 75 m

The breadth of the wall = 32 m

Area of the wall = length x breadth = 75 m x 32 m = 2400 sq. m

For 1 sq. m of painting costs Rs 3

Thus, for 2400 sq. m, the cost of painting the wall will be = 3 x 2400 = Rs 7200

Example 5: A floor whose length and width is 50 m and 40 m respectively needs to be covered by rectangular tiles. The dimension of each tile is 1 m x 2 m. Find the total number of tiles that would be required to fully cover the floor.

Solution:

Length of the floor = 50 m

The breadth of the floor = 40 m

Area of the floor = length x breadth = 50 m x 40 m = 2000 sq. m

Length of one tile = 2 m

The breadth of one tile = 1 m

Area of one tile = length x breadth = 2 m x 1 m = 2 sq. m

No. of tiles required = area of floor/area of a tile = 2000/2 = 1000 tiles

Frequently Asked Questions on Area of Rectangle

The formula of the area of a rectangle is used to find the area occupied by the rectangle within its boundary. In the above example, the area of the rectangle whose length is 4 inches and the width is 3 inches is 12 square inches. We have 4 × 3 = 12. The area of a rectangle is obtained by multiplying its length and width. Thus, the formula for the area, ‘A’ of a rectangle whose length and width are ‘l’ and ‘w’ respectively is the product “l × w“.

Area of a Rectangle = (Length × Breadth) square units

The area of a rectangle is equal to its length times its width. Follow the steps mentioned below to find the area of a rectangle:

• Step 1: Note the dimensions of length and breadth from the given data.
• Step 2: Find the product of length and breadth values.
• Step 3: Give the answer in square units.

Let us take an example to understand the calculation of the area of a rectangle. We will find the area of the rectangle whose length is 15 units and breadth is 4 units. To find the area, first, find the length and width.

Given, length = 15 units and width = 4 units.

The formula to find the area of a rectangle is A = l × w. Substitute 15 for ‘l’ and 4 for ‘w’ in this formula. This implies, area of the rectangle = 15 × 4 = 60.

Therefore, the area of the rectangle = 60 square units.

The diagonal of a rectangle is the straight line inside the rectangle connecting its opposite vertices. There are two diagonals in the rectangle and both are of equal length. We can find the diagonal of a rectangle by using the Pythagoras theorem.

(Diagonal)² = (Length)² + (Breadth)²

(Length)² = (Diagonal)² – (Breadth)²

Length = ⎷(Diagonal)² – (Breadth)²

Now, the formula to calculate the area of a rectangle is Length × Breadth. Alternatively, we can write this formula as ⎷((Diagonal)² – (Breadth)²) × Breadth.

So, Area of a Rectangle = Breadth (⎷(Diagonal)² – (Breadth)²).

Have you ever wondered why the formula to find the area of a rectangle is Length × Breadth? Let’s derive the formula of the area of a rectangle. We will draw a diagonal AC in the rectangle ABCD. Clearly, the diagonal AC divides the rectangle ABCD into two congruent triangles. The area of the rectangle is the sum of the area of these two triangles.

Area of Rectangle ABCD = Area of Triangle ABC + Area of Triangle ADC

= 2 × Area of Triangle ABC

= 2 × (1/2 × Base × Height)

= AB × BC

= Length × Breadth

Area of Rectangle Examples

Example 2:

Peter reads his favorite comic book in his study room. Can you

A rectangle is a closed figure which has four sides and the angle formed by adjacent sides is 90°. A rectangle can have a wide range of properties. Some of the important properties of a rectangle are given below.

• A rectangle is a quadrilateral.
• The opposite sides of a rectangle are equal and parallel to each other.
• The interior angle of a rectangle at each vertex is 90°.
• The sum of all interior angles is 360°.
• The diagonals bisect each other.
• The length of the diagonals is equal.
• The length of the diagonals can be obtained using the Pythagoras theorem. The length of the diagonal with sides a and b is, diagonal = √( a² + b²).
• Since the sides of a rectangle are parallel, it is also called a parallelogram.
• All rectangles are parallelograms but all parallelograms are not rectangles.

The diagonal of a rectangle is a line segment that joins any two of its non-adjacent vertices. In the following rectangle, AC and BD are the diagonals that are equal in length. A diagonal cuts a rectangle into 2 right-angled triangles, in which the diagonal forms the hypotenuse, and the two adjacent sides of the rectangle form the other two sides of the triangle.

Diagonal of Rectangle Formula

The formula for the diagonal of a rectangle is derived using the Pythagoras theorem. Following the figure given above, let us consider a rectangle of length ‘l’ and width ‘w’. Let the length of each diagonal be “d”. Applying Pythagoras theorem to the triangle ABD, d² = l² + w². Taking square root on both sides, √(d²) = √( l² + w²). Thus, the diagonal of a rectangle formula is diagonal (d): √(l² + w²) and thus the diagonals of a rectangle can be calculated when the length and width of the rectangle are known.

Diagonal of Rectangle (d) = √(l² + w²)

Perimeter of Rectangle Formula

The formula for the perimeter, ‘P’ of a rectangle whose length and width are ‘l’ and ‘w’ respectively is 2(l + w).

Perimeter of a Rectangle Formula = 2 (Length + Width)

Types of Rectangles

A quadrilateral whose opposite sides are equal and adjacent sides meets at 90° is called a rectangle. A rectangle has two equal diagonals. The length of the diagonals is calculated by using the length and width. There are two types of rectangles:

• Square
• Golden Rectangle

Golden Rectangle

A golden rectangle is a rectangle whose ‘length to the width’ ratio is similar to the golden ratio, 1: (1+⎷5)/2. Its sides are defined according to the golden ratio, that is, 1: 1.618. For instance, if the width is about 1 foot long then the length will be 1.168 feet long.

Examples on Properties of Rectangle

• Example 2: Elsa wants to build a rectangular fence for her garden. The perimeter of the fence is 30 feet. The length of the fence is 10 feet. Can you help Elsa find the width of the fence?

  Solution:

  We know that the formula to calculate the perimeter of a rectangle is, Perimeter of a Rectangle = 2 (Length + Width). We have perimeter = 30 feet and length = 10 feet, So, let us find the width using the perimeter formula. Let us substitute the known values in the formula, Perimeter of a Rectangle = 2 (Length + Width), 30 = 2 (10 + width). Solving it further, we get, 10 + width = 15, and width = 5 units

  Therefore, the width of the fence = 5 feet

• Example 3: State true or false:

  a.) The opposite sides of a rectangle are equal and parallel to each other.

  b.) The length of the diagonals can be obtained using the Pythagoras theorem.

  c.) All parallelograms are rectangles.

  Solution:

  a.) True, the opposite sides of a rectangle are equal and parallel to each other.

  b.) True, the length of the diagonals can be obtained using the Pythagoras theorem.

  c.) False, all rectangles are parallelograms but all parallelograms are not rectangles.

Why is a Rectangle Not a Regular Polygon?

What is the Formula for the Area of a Rectangle?

Important Rectangle Formulas

The formula of the area and perimeter of the rectangle is given below.

Given that ‘l’ represents the length and ‘b’ represents the breadth of the rectangle, then the

Derive and Calculate the Area of a Rectangle

To derive the area of a rectangle, we use the unit squares. Divide the rectangle MNOP into unit squares. The area of a rectangle MNOP is the total number of unit squares in it.

Now, finding the area of a rectangle

Rectangle MNOP in unit squares each with 1 sq. inch

Therefore, the total area of the rectangle MNOP is equal to 48 sq. inch.

Also, following the perspective, we discovered that the area of a rectangle is mandatorily the product of its two sides. Here, the length of MN is 6 inches and the breadth of NO is 8 inches. The area of MNOP is the product of 6 and 8, which is 48.

Instead, formulas to find the area of a rectangle is derived by dividing the figure into two equal-sized right triangles. For example, in the given rectangle MNOP, a diagonal from the vertex M is drawn to O.

The diagonal MO divides the rectangle into 2 equivalent right-angle triangles.

Therefore, the area of MNOP will be:

   ⇒ Area (MNOP) = Area (MNO) + Area (MPO)

   ⇒ Area (MNOP) = 2 × Area (MNO)

   ⇒ Area (MNO) = ½ × base × height

   ⇒ Area (MNOP) = 2 × (½ × b × h)

    ⇒ Area (MNOP) = b × h

Solved Examples

Here, we will perform practical problems using step-by-step solutions in order to find the area of a rectangle as well as other measurements.

Example:

Calculate the area and the perimeter of the rectangle box that measures 15 cm in length and 8cm in breadth.

Solution:

Given:

Length = 15 cm,

Breadth = 8 cm

Now applying the area of rectangle formula i.e. = length × breadth

= (15 × 8) cm²

= 120 cm²

Using the formula for Perimeter of rectangle = 2 (length + breadth)

 = 2 (15 + 8) cm

= 2 × 23 cm

= 46 cm

Example:

Evaluate the breadth of the Television set rectangular in shape whose area is 320 m2 and whose length is 40 m. Find its perimeter.

Solution:

We know that the breadth of the rectangular television set = Area/length

= 320m/40m

= 8 m

Thus, the perimeter of the rectangular television set = 2 (length + breadth)

= 2(40 + 8) m

 = 2 × 48 m

= 96 m

Sample Questions

Question 1: Find the area of the rectangle whose length is 21 units, width is 11 units.

Solution:

Given, length = 21 units and width = 11 units.

The formula to observe the area of a rectangle is A = length × breadth (l × b).

Substitute 21 for ‘l’ and11 for ‘w’ in this equation.

So, area of the rectangle = 21 × 11 = 231 sq units.

Question 2: Find the area of a rectangle of length of 12 mm and breadth of 8 mm.

Solution:

Length of a rectangle = 12 mm.

Breadth of a rectangle = 8 mm.

Area of a rectangle = length × breadth

= 12 × 8 sq mm.

= 96 sq mm.

Question 3: Finding the area of a rectangle whose length is 10.5 cm and breadth is 5.5 cm.

Solution:

Length of the rectangle (l) = 10.5 cm

Breadth of the rectangle (b) = 5.5 cm

Area of a rectangle = length × breadth (l × b)

Area of the rectangle = 10.5 × 5.5

= 57.75 cm².

Question 4: The area of a rectangle is 32 cm². If its breadth is 4 cm then find its length.

Solution:

Area of rectangle = 32 cm²

Breadth of rectangle = 4 cm

Length of rectangle = Area of the rectangle/Breadth of the rectangle

= 32 cm²/4 cm

= 8 cm.

So, the length of the rectangle is 8 cm.

Question 5: Find the perimeter of a rectangle whose length and width are 11 cm and 5.5 cm, respectively.

Solution:

Length = 11 cm and Width = 5.5 cm

The perimeter of a rectangle = 2(length + width)

Substitute the value of length and width here,

Perimeter, P = 2(11 + 5.5) cm

P = 2 × 16.5 cm

Therefore, the perimeter of a rectangle = 33 cm.

Question 6: A rectangular yard has a length equal to 12 cm and a perimeter equal to 60 cm. Find its width.

Solution: 

Perimeter = 60 cm

Length = 10 cm

Let W be the width.

From the formula,

Perimeter, P = 2(length + width)

Substituting the values,

60 = 2(12 + width)

12 + W = 30

W = 30 – 12 = 18

Hence, the width is 20cm.

Question 7: Find the perimeter of a rectangle whose length and width are 12cm and 4cm, respectively.

Solution:

Given,

Length = 12cm

Width = 4 cm

Perimeter of Rectangle = 2(Length + Width)

= 2(12 + 4) cm

= 2 × 16 cm

Therefore, the perimeter of a rectangle = 32 cm.

Question 8: Find the perimeter of a rectangle whose length is 21 cm and width is 13 cm.

Solution:

Given,

Length = 21cm

Width = 13 cm

Perimeter of Rectangle = 2(Length + Width)

= 2(21 + 13) cm

= 2 × 34 cm

Therefore, the perimeter of a rectangle = 68cm.

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