In geometry, the area of a plane 2D shape is the region covered by it (region bounded by the perimeter) in a two-dimensional plane. In other words, we can say that area of any shape is the number of unit squares that can fit into it. Here a unit square refers to a square of side $1$ unit. One good example is graph paper. By counting the number of squares in a region, you can find the area of that region.

## Rectangle – A 2D Plane Figure

A rectangle is a 2D shape, whose opposite sides are equal and parallel. The measure of all four angles of a rectangle is equal and it is $90^{\circ}$.

## What is the Area of Rectangle?

The area of a rectangle is the region occupied by it within its four sides or boundaries. <image>

Let’s divide the rectangle into a number of small squares each of length $1$ unit. This unit can be $mm$, $cm$, $in$, $m$, etc.

Now, each small square is of area $1 sq unit$ or $1 unit^{2}$. The area of the rectangle will be the total number of small squares present in it.

In the case of the above rectangle, the area is $24$ sq units or $24 unit^{2}$.

### Usage of Area of a Rectangle

The area of a rectangle makes things easier and helps us in calculating the space covered in our day-to-day lives.

For example,

- If you need to cover a notebook, you can easily calculate how much paper you would need by finding the area.
- If a farmer needs to plough his field, the area of the field will give him the exact region he would need.
- For the construction plan of the house, we need to know the number of tiles needed for flooring that is possible by the area formula

### Area of Rectangle Formula Using Sides

The formula for area is equal to the product of the length and width (or breadth) of the rectangle. That is if $l$ is the length and $w$ is the width of a rectangle, then its area is given by $A = l \times w$. For example, if the length and width of a rectangle are $4 cm$ and $3 cm$ respectively, then the area of rectangle will be $4 \times 3 = 12 cm^{2}$ or $12 sq cm$.

**Note:** Area is always represented in terms of $sq unit$ or $unit^{2}$, such as $sq cm$ or $cm^{2}$.

### How to Calculate Area of Rectangle Using Sides?

The area of a rectangle is equal to its length times its width. The following steps are used to find the area of a rectangle:

**Step 1:** Note down the length and width of a rectangle

**Step 2:** Convert the length and width to the same units, if units of length and width are not the same

**Step 3:** Substitute the values of length and width of a rectangle in the formula

**Step 4:** Simplify the expression in the formula to get the area in square unit

### Examples

**Ex 1:** Find the area of a rectangle, if its length is $14 in$ and width is $11 in$

Length of rectangle $l = 14 in$

Width of rectangle $w = 11 cm$

Area of a rectangle $A = l \times w$

Substitute the values of $l$ and $w$ in the formula.

$A = 14 \times 11 = 154 in^{2}$

Therefore, the area of a rectangle of length and width $14 in$ and $11 in$ is $154 in^{2}$.

**Ex 2:** Find the area of a rectangle whose length is $1.5 m$ and width is $60 cm$.

Length of rectangle $l = 1.5 m$

Width of rectangle $w = 60 cm$

Observe that the length and width do not have the same unit. So, you’ve to convert either $m$ to $cm$ or $cm$ to $m$.

Let’s convert the width is $m$.

Width $w = 60 cm = \frac {60}{100} = 0.60 m$.

Area of a rectangle $A = l \times w$

Substitute the values of $l$ and $w$ in the formula.

$A = 1.5 \times 0.60 = 0.9 m^{2}$

Therefore, the area of a rectangle of length and width $1.5 m$ and $60 cm$ is $0.9 m^{2}$.

You can also find the area by converting the unit of length into centimetre.

Length = $1.5 m = 1.5 \times 100 = 150 cm$.

Area of a rectangle $A = l \times w$

Substitute the values of $l$ and $w$ in the formula.

$A = 150 \times 60 = 9000 cm^{2}$

Therefore, the area of a rectangle of length and width $1.5 m$ and $60 cm$ is $9000 cm^{2}$.

**Note:** The areas $9.9 m^{2}$ and $9000 cm^{2}$ are the same. $0.9 m^{2} = 0.9 \times 10000 = 9000 cm^{2}$.

**Ex 3:** Manoj wants to polish the table of length $5 ft$ and width $4 ft$. If the cost of polishing is ₹$50$ per $ft^{2}$, what is the cost of polishing?

Length of rectangular table $l = 5 ft$

Width of rectangular table $w = 4 ft$

The region of the table to be polished is the same as the area of the table.

Area of a rectangle $A = l \times w$

Substitute the values of $l$ and $w$ in the formula.

$A = 5 \times 4 = 20 ft^{2}$

Therefore, the cost of polishing the table is $20 \times 50 =$₹ $1000$.

**Ex 4:** How many pieces of stone slabs each $26 cm$ long and $10 cm$ broad will be required to lay a path $260 m$ long and $15 m$ wide?

Dimensions of each square stone slab are $l = 26 cm$ and $w = 10 cm$

Therefore, the area of each stone slab = $26 \times 10 = 260 cm^{2}$.

Dimensions of the path are $L = 260 m = 260 \times 100 = 26000 cm$ and $W = 15 m = 15 \times 100 = 1500 cm$

Therefore, the area of path = $26000 \times 1500 = 39,000,000 cm^{2}$

The number of stone slabs required = $\frac {\text{Area of path}}{\text{Area of one stone slab}} = \frac {39,000,000}{260} = 150000$.

### Area of a Rectangle Formula Using a Diagonal

You can also find the area of a rectangle if the lengths of its diagonal and one of its sides (length or width) are known. In this case, you use the Pythagoras Theorem to find the area of a rectangle.

Let’s consider a rectangle of length $l$, width $w$ and diagonal $d$.

By Pythagoras Theorem, we get $d^{2} = l^{2} + w^{2}$

The above statement can be written as

- $w^{2} = d^{2} – l^{2} => w = \sqrt{d^{2} – l^{2}}$
- $l^{2} = d^{2} – w^{2} => l = \sqrt{d^{2} – w^{2}}$

Therefore, the area of rectangle is

- when diagonal and length of a rectangle are known: $A = l \times \sqrt{d^{2} – l^{2}}$
- when diagonal and width of a rectangle are known: $A = w \times \sqrt{d^{2} – w^{2}}$

### Examples

**Ex 1:** Find the area of a rectangle whose length is $40 cm$ and diagonal is $50 cm$.

Length of rectangle $l = 40 cm$

Diagonal of rectangle $d = 50 cm$

Area of rectangle $A = l \times \sqrt{d^{2} – l^{2}}$

Substitute the values of $l$ and $d$ in the formula.

$A = 40 \times \sqrt{50^{2} – 40^{2}} = 40 \times \sqrt{2500 – 1600} = 40 \times \sqrt{900} 40 \times 30 = 1200 cm^{2}$.

Therefore, the area of a rectangle of length and diagonal $40 cm$ and $50 cm$ is $1200 cm^{2}$.

**Ex 2:** Find the area of a rectangle whose width is $5 ft$ and diagonal is $13 ft$.

Width of rectangle $w = 5 ft$

Diagonal of rectangle $d = 13 ft$

Area of rectangle $A = w \times \sqrt{d^{2} – w^{2}}$

Substitute the values of $w$ and $d$ in the formula.

$A = 5 \times \sqrt{13^{2} – 5^{2}} = 5 \times \sqrt{169 – 25} = 5 \times \sqrt{144} = 5 \times 12 = 60 ft^{2}$

Therefore, the area of a rectangle of width and diagonal $5 ft$ and $13 ft$ is $60 ft^{2}$.

## Conclusion

The area of a 2D object is the region bounded by its perimeter in the space. The area is measured in a square unit or $unit^{2}$. The area of a rectangle depends on its length and width and is calculated by calculating the product of length and width.

## Practice Problems

- Find the area of a rectangle whose length and width are
- Length = $9 unit$ and Width = $7 unit$
- Length = $15 mm$ and Width = $12 mm$
- Length = $17 in$ and Width = $8 unit$

- Find the length of the rectangle whose area and width are
- Area = $96 cm^{2}$ and width = $6 cm$
- Area = $144 m^{2}$ and width = $8 m$

- Find the width of the rectangle whose area and length are
- Area = $312 m^{2}$ and width = $13 m$
- Area = $126 mm^{2}$ and width = $7 cm$

- The area of a rectangular fence is 500 square feet. If the width of the fence is 20 feet, then find its length
- Find the area of a rectangle whose length of $12 cm$ and the width $3$ times smaller
- How many squares with the side of $2 cm$ cover the surface of a rectangle with a length of $24$ cm and a width of $8 cm$?

## Recommended Reading

- What is Length? (With Definition, Unit & Conversion)
- Weight – Definition, Unit & Conversion
- What is Capacity (Definition, Units & Examples)
- What is Time? (With Definition, Facts & Examples)
- What is Temperature? (With Definition & Units)
- Reading A Calendar
- Perimeter of Rectangle – Definition, Formula & Examples
- Perimeter of Square – Definition, Formula & Examples
- Area of Square – Definition, Formula & Examples

## FAQs

### What is a rectangle?

A rectangle is a closed two-dimensional figure with four sides where opposite sides are equal and parallel to each other. The rectangle shape has all the angles equal to $90^{\circ}$.

### What is the formula for the area of a rectangle?

The area of a rectangle $\left(A \right)$ is the product of its length $l$ and width or breadth $w$’. So, Area of rectangle = $\left(l \times w \right)$ square units.

### How can we find the area of a rectangle using its diagonal?

We can find the diagonal of a rectangle by using the Pythagoras theorem: $d^{2} = l^{2} + w^{2}$, where $l$, $w$, and $d$ are the length, width and the diagonal of a rectangle respectively. Now, the formula to calculate the area of a rectangle becomes

a) when the diagonal and length of a rectangle are known: $A = l \times \sqrt{d^{2} – l^{2}}$

b) when the diagonal and width of a rectangle are known: $A = w \times \sqrt{d^{2} – w^{2}}$

### What do you mean by the area of a rectangle?

The area of a rectangle is the region occupied by the perimeter (boundary) of the rectangle.