Area of Rectangle: A rectangle is a 2-D plane figure that has 4 sides and 4 internal angles. In a rectangle, the opposite sides and angles are the same in value. A rectangle is defined as a quadrilateral figure having four sides and the opposite sides are similar in length and parallel to one another. In our day-to-day life, we see several rectangular-designed stuff like tables, books, boxes, mobile phones, walls, television, beds, almirah, etc. In this article, we provide you with the properties of a rectangle, the area of a rectangle, its formulas, and some relevant question answers for a better understanding of the concepts of the rectangle.
Area of Rectangle
In the study of geometry, the area of a rectangle is defined as the region enveloped by the rectangle in a 2-dimensional plane. All four internal angles of a rectangle are right angles (90°). The surface area extended by a rectangle depends on its four sides. The formula for the area of a rectangle is equivalent to the multiplication of the length and breadth of the rectangle. The rectangular region covered by the perimeter of the rectangle is the area of a rectangle.
Area of Rectangle Formula
The rectangle area is the region covered inside the outer boundary of the rectangle made by its four sides. The area of a rectangle can be enumerated by the product of its breadth and its length.
Let the length of a rectangle is ‘L’ and its breadth or width be ‘B’ and the Area of a Rectangle be indicated by ‘A’. Then the formula to find out the rectangle area will be the product of length and breadth. The area of any figure like a rectangle or else is always represented in square units.
Area of Rectangle= Length × Width
A = L × B (in square units)
Area of Rectangle Formula’s Proof
The diagonals of a rectangular shape bisect the shape into 2 similar right-angled triangles. Hence, the rectangle area is equivalent to the addition of the area of two right-angled triangles.
Let ABCD is a rectangular shape.
Now considering diagonal AC bisects the rectangle into two right-angled triangles like ∆ABC and ∆ADC.
As we know that ∆ABC and ∆ADC are identical right-angled triangles.
Area of ∆ABC = ½ x Base of ∆ABC x Height of ∆ABC = ½ x AB x BC = ½ x B x L ……….1
Area of ∆ADC = ½ x Base of ∆ADC x Height of ∆ADC = ½ x CD x AD = ½ x B x L ……….2
Since Area of Rectangle ABCD = Area of ∆ABC + Area of ∆ADC
From the 1st and 2nd expressions, we have
Area of ABCD Rectangle = 2(½ x B x L)
Area of ABCD Rectangle = L x B
Hence proved that the Area of a Rectangle is = Length x Breadth (Width)
Area of Rectangle Using Diagonals
The rectangle area can be calculated with the help of its diagonals as given below.
By applying the Pythagoras Theorem, we get
(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
As we know that the rectangle area is the multiplication of length and width
Area of Rectangle = Length × Width
By putting the 2nd equation of width, we get
Area of Rectangle = Length × √(Diagonal)² – (Length)²
By putting the 1st equation of length, we get
Area of Rectangle = √ (Diagonal)² – (Width)² × width
Rectangle Properties
There are many properties of a rectangle 2-D plane figure which is used in geometry. The key properties of a rectangle are mentioned below.
- A rectangle is a kind of quadrilateral-shaped figure.
- The opposite sides of a rectangle are similar in length and parallel to one another.
- The four interior angles of a rectangle have values of 90° at each vertex.
- The addition of all four interior angles of a rectangle results in 360° (90°+90°+90°+90°).
- The rectangle diagonals bisect one another.
- The two diagonals of a rectangle are the same in terms of their length.
- 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).
- When all four sides of a rectangle are parallel in nature then it is called a parallelogram.
- 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 of that rectangle is 7 cm. Calculate the area of that rectangle.
Solution: Given that Length L = 12 cm and Breadth B = 7 cm
As we know the Area of Rectangle A = L × B
By substituting the values of L and B in the above formula, we get
A = 12 × 7
A = 84 cm²
Question 2: The width of a table is 15 m and the area of that table is 135 m². Find out the length of that table.
Solution: Given that Width of the table = 15 m
and the Area of the table = 135 m²
As we know that table is generally rectangular-shaped so the area of the rectangle will be applied here.
Area of Rectangle = Length × Breadth
A = L × B
135 = L x 15
Length of the table L = 135 / 15 = 9 m
Question 3: Calculate the surface area of a rectangular smartboard in square meters whose length is 200 cm and width is 120 cm.
Solution: First convert the unit from cm to m, 1 meter = 100 cm
Length of the smartboard = 200 cm = 2 m
Breadth of the smartboard = 120 cm = 1.2 m
Area of the smartboard = Area of a Rectangle = Length x Width = 2 m x 1.2 m = 2.4 metres²
Question 4: The length and width of a rectangular wall are given with values of 30 m and 15 m respectively. Calculate the painting cost of that wall if the painting rate is Rs 2 per sq. m.
Solution: Length of the rectangular wall = 30 m
The width of the rectangular wall = 15 m
Area of that rectangular wall = Length of wall x Width of wall = 30 m x 15 m = 450 m²
The painting cost for the coverage of 1 m² is Rs 3
Hence the painting cost for the coverage of 2400 m² will be = 3 x 450 = Rs 1350
Question 5: The length of a book is 5 cm. Its area is 20 sq. cm. What will be its width?
Solution: Given that the Area of the book = 20 sq. cm.
Length of the book = 5 cm
As we know the Area of a rectangle is = Length x Width
Hence, the width can be calculated by = Area / Length
So width of the book = 20/5 = 4 cm