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.
