Area of Rectangle

In 2D Geometry, a Rectangle is a fundamental shape consisting of four sides, 4 edges, and one diagonal, with all the sides intersecting at a right angle or 90° angle and opposite sides parallel and equal to each other. The rectangle of 2D geometry makes the foundation for understanding the concept of 3D geometry. 

1.0Introduction to Area of Rectangle

The area of the rectangle is the measure of the amount of space enclosed by its four sides. Finding the area of the rectangle in two-dimensional geometry is a basic yet important concept in mathematics as well as real-life problems. Whether you're planning a room layout, designing a garden, or solving math problems, understanding how to calculate the area is essential. 

The formula to calculate the Area of the Rectangle is: 

$\text{Area of rectangle}=Length(l)\times Breadth(b)$

Here, 

• Length (l) is the longer side of a rectangle.
• Breadth (b) is the shorter side of the rectangle. 

Generally, the Unit of the area of a rectangle depends upon the units used in the length and breadth of the rectangle. The area is measured in square units as square centimetres cm², Square meters m², square inches in², etc. 

Note: Make sure that the units of both length and breadth are in the same unit to avoid any blunder while calculating the area. 

How to Find the Area of the Rectangle

The area of the rectangle can be found in the three easy steps: 

Step 1: Determine the length and breadth of the rectangle. 

Step 2: Convert the units of both length and breadth into the same units if not given already. 

Step 3: Multiply the length and breadth and write the result in the square units of the rectangle. 

Perimeter of Rectangle 

The perimeter and area of a rectangle help us in calculating two different aspects of a rectangle, where the area measures the space with the enclosed sides of the rectangle and the perimeter measures the total distance of the enclosed sides. In simple words, the perimeter is the sum of all sides of a rectangle. The formula for the Perimeter of a Rectangle is: 

Perimeter=2(Length+Breath)

Diagonal of Rectangle

A diagonal is a line connecting two opposite edges of a rectangle, cutting it into two equal right-angled triangles. The diagonal gives the measure of the stretch of a rectangle across its space. The diagonal of a rectangle can be measured by the following formula: 

$\text{Diagonal}=\sqrt{(\text{length}{)}^2+(\text{breadth}{)}^2}$

2.0Area of Rectangle and Triangle

The diagonal of a rectangle cuts the rectangle into two equal areas of Right angle triangles. This means that the area of the rectangle in the context of the area of the triangle can be written as: 

Area of Rectangle =

$2\times \text{Area of the Triangle}$ …….(1)

Here $\text{Area of the Triangle}=\frac{1}{2}\times Base\times Height$ ……..(2)

From equation 1 and 2

$\text{Area of Rectangle}=2\times (\frac{1}{2}\times Base\times Height)=Base\times Height$

Hence, this relation also gives proof for the area of the rectangle. 

3.0Solved Problems: 

Problem 1: The diagonal of a rectangle measures 10 cm. If the length of the rectangle is 6 cm, calculate the area of the rectangle.

Solution:

The diagonal of the rectangle is 10cm, and the length of the rectangle is 6cm

$10=\sqrt{(\text{(6)}{)}^2+(\text{breadth}{)}^2}$

Squaring both sides: 

$10^2=36+b^2$

$b^2=100-36$

$b=\sqrt{64}=8cm$

And $\text{Area of rectangle}=Length(l)\times Breadth(b)$

$\text{Area of rectangle}=68=48c{m}^2$

Problem 2: The perimeter of a rectangle is 30 cm. The length of the rectangle is 10 cm. Calculate the area of the rectangle.

Solution:

length = 10, perimeter = 30, so: 

Perimeter=2(Length+Breath)

30=2(10+b)

15=10+b

b=5cm

And $\text{Area of rectangle}=Length(l)\times Breadth(b)$

$\text{Area of rectangle}=105=50c{m}^2$

Problem 3: A company sells rectangular tiles that measure 20 cm by 30 cm. If the cost of one tile is Rs. 100, calculate how much it would cost to cover a rectangular floor that is 4 meters long and 3 meters wide.

Solution: length of floor = 4m = 400cm 

Breadth of the floor = 3m = 300cm 

Area of the floor = lb = 400300 = 1,20,000 $c{m}^2$

Area of one tile = lb = 2030 = 600 $c{m}^2$

Number of tiles used in the floor = $\frac{120000}{600}$ = 200 tiles

Total cost of tiles = $200\times 100$ =Rs 20,000