Area of a Rectangle: Formula, Definition, and Solved Examples

Area of a Rectangle: A rectangle is a 2D geometric shape featuring four sides and four internal right angles. In a rectangle, opposite sides are equal in length and parallel to one another. We encounter rectangular objects daily, such as tables, books, mobile phones, walls, and beds. This article explores the core properties of a rectangle, the mathematical formula for its area, and provides practical examples to help you master the concept.

Area of Rectangle

In geometry, the area of a rectangle is defined as the total region enclosed by its four sides on a 2-dimensional plane. Every internal angle of a rectangle measures exactly 90°. The surface area depends entirely on the dimensions of these sides; specifically, the area is calculated by multiplying the length by the width. The area represents the flat space contained within the perimeter boundary. 

Area of Rectangle Formula

The area of a rectangle is the space contained within its four-sided outer boundary. You can calculate the area of a rectangle by finding the product of its length and breadth. 

Let the length of a rectangle be 'L', its breadth or width be 'B', and the area be represented by 'A'. To determine the area, simply multiply the length by the breadth. The resulting area of any such 2D figure is always expressed in square units.

Area of a Rectangle = Length × Width

A = L × B (in square units)

Area of Rectangle Formula’s Proof

The diagonals of a rectangle bisect the shape into two congruent right-angled triangles. Consequently, the total area of the rectangle is equal to the sum of the areas of these two identical triangles.

Consider rectangle ABCD. 

If we draw a diagonal AC, it splits the rectangle into two right-angled triangles: ΔABC and ΔADC.

Both ΔABC and ΔADC are identical right-angled triangles.

Area of ΔABC = ½ × Base × Height = ½ × AB × BC = ½ × B × L ……1

Area of ΔADC = ½ × Base × Height = ½ × CD × AD = ½ × B × L ……2

Since the area of rectangle ABCD = Area of ΔABC + Area of ΔADC

Using the calculations from the 1st and 2nd expressions:

Area of Rectangle ABCD = 2 × (½ × B × L)

Area of Rectangle ABCD = L × B 

This proves that the Area of a Rectangle = Length × Breadth (Width)

Area of Rectangle Using Diagonals

The area of a rectangle can also be determined using its diagonal measurement as shown below.

Applying the Pythagorean theorem, we have:

(Diagonal)² = (Length)² + (Width)²

(Length)² = (Diagonal)² – (Width)²

Length = √(Diagonal)² – (Width)² ……1

(Width)² = (Diagonal)² – (Length)²

Width = √(Diagonal)² – (Length)² ……2

Since the area is the product of length and width:

Area of Rectangle = Length × Width

By substituting the second equation for width, we get:

Area of Rectangle = Length × √(Diagonal)² – (Length)²

By substituting the first equation for length, we get: 

Area of Rectangle = √(Diagonal)² – (Width)² × Width

Rectangle Properties

There are several geometric properties of a rectangle. Key characteristics are highlighted below.

1. A rectangle is a kind of quadrilateral-shaped figure.
2. The opposite sides of a rectangle are similar in length and parallel to one another.
3. The four interior angles of a rectangle have values of 90° at each vertex.
4. The addition of all four interior angles of a rectangle results in 360° (90°+90°+90°+90°).
5. The rectangle diagonals bisect one another.
6. The two diagonals of a rectangle are the same in terms of their length.
7. The diagonal length can be calculated by using the Pythagoras Theorem. If the diagonal length with sides a and b then the length of the diagonal is = √( a² + b²).
8. When all four sides of a rectangle are parallel in nature then it is called a parallelogram.
9. It is to be noted here that all rectangles are considered parallelograms surely but all parallelograms can not be called rectangles.

Area of Rectangle Solved Questions

Question 1: If the length of a rectangle is 12 cm and the breadth is 7 cm, calculate the area.

Solution: Given: Length (L) = 12 cm and Breadth (B) = 7 cm.

Using the formula Area (A) = L × B

Substituting the values, we get: 

A = 12 × 7

A = 84 cm²

Question 2: The width of a table is 15 m and its area is 135 m². Find its length.

Solution: Width = 15 m

Area = 135 m² 

Assuming the table is rectangular, we use the rectangle area formula.

Area = Length × Breadth

A = L × B

135 = L × 15

Length (L) = 135 / 15 = 9 m

Question 3: Calculate the area of a rectangular smartboard (length 200 cm, width 120 cm) in square meters.

Solution: Convert units to meters (1 m = 100 cm).

Length = 200 cm = 2 m

Breadth = 120 cm = 1.2 m

Area = Length × Width = 2 m × 1.2 m = 2.4 m² 

Question 4: A wall is 30 m long and 15 m wide. Calculate the painting cost if the rate is Rs 3 per m².

Solution: Length = 30 m

Width = 15 m

Area = 30 m × 15 m = 450 m²

Painting cost per m² = Rs 3

Total cost = 3 × 450 = Rs 1350

Question 5: A book has a length of 5 cm and an area of 20 cm². What is its width?

Solution: Area = 20 cm²

Length = 5 cm

Since Area = Length × Width

Width = Area / Length

Width = 20 / 5 = 4 cm