What is a rectangle?
A rectangle is a four sided quadrilateral, having equal opposite sides and all the interior angles are equal to 90 degrees. A door, a curtain or even a book, are a few examples of objects that are rectangular in shape.
The area of a rectangle is the measure of space occupied inside the boundary of the rectangle. In other words, the amount of space covered by the rectangle in a two dimensional plane is its area.
Area of a rectangle formulas
The area of a rectangle is given by the following formula,
Area of a rectangle, A = (l x w)
Where,
A \(\rightarrow\) area of rectangle
l \(\rightarrow\) length of rectangle
w \(\rightarrow\) width of rectangle
Derivation of the formulas
The product of the length and the width of the rectangle gives the area of the rectangle.
Area of a rectangle, A = length of the rectangle x width of the rectangle
Area of a rectangle, A = l x w
Area is measured in square units. If the width and length of the rectangle is in meters (m), then the area will be measured in square meters \((m^2)\).
For example, if the length of a rectangle is 4 cm and the width is 8 cm then its area is given by:
Area of a rectangle, A = l x w
= 8 x 4
= 32 \(cm^2\)
Solved Examples
Example 1:
What is the area of the given rectangle?
Solution:
Area of a rectangle, A = l x w [Area of rectangle formula]
= 75 [Substitute values]
= 35 \(cm^2\) [Multiply]
Therefore, the area of the given rectangle is 35 \(cm^2\)
Example 2:
What is the area of the given rectangle?
Solution:
Area of a rectangle, A = l x w [Area of rectangle formula]
= 12 x 13 [Substitute values]
= 156 [Multiply]
Therefore, the area of the given rectangle is 156 square inches.
Example 3:
The area of a rectangle is 72 \(cm^2\) . What is the width of the rectangle if the length of the rectangle is 9 cm?
Solution:
If the area of the rectangle is 72 \(cm^2\) and its length is 9cm, we can use the formula for the area of a rectangle to find the width of the rectangle.
Area of a rectangle, A = l x w [Area of rectangle formula]
72 = 9 x w [Substitute values]
\(\frac{72}{9}\) = w [Divide both sides by 9]
8 = w [Simplify]
Hence the width of the given rectangle is 8 cm.
Example 4:
Find the area of the rectangle shown below. What can be the different possible dimensions of a rectangle with the same area.
Solution:
Area of a rectangle, A = l x w [Area of rectangle formula]
= 12 cm x 2 cm [Substitute values]
= 24 \(cm^2\) [Multiply]
Alternate dimensions of a rectangle with area = 24 \(cm^2\)
= 6 cm x 4 cm
= 8 cm x 3 cm
= 24 cm x 1 cm
Hence, the alternate dimensions of rectangle with area 24 \(cm^2\) are 6 cm, 4 cm and 8 cm, 3 cm and 24 cm, 1 cm.
Example 5:
Lance and Mark want to buy a carpet that would stretch beyond the area of the study table by 10 m2. What should be the area of the carpet if the dimensions of the study table are given below?
Solution:
The area of the carpet can be obtained by adding the area of the study table and 10 \(m^2\) .
Area of the study table, A = l x w [Area of rectangle formula]
= 8m x 5m [Substitute values]
= 40 \(m^2\) [Multiply]
Area of the carpet = 40 + 10 = 50
Therefore, the area of the carpet should be 50 \(m^2\)
Example 6:
A rectangular piece of wood is divided into two equal portions lengthwise. If the dimensions of one of the portions are as follows, what is the total area of the entire section?
Solution:
The two portions are equal and divided lengthwise. So the length of the entire section will be twice the length of each section, that is,
= 2 x 4m
= 8m
Area of entire section, A = l x w [Area of rectangle formula]
A = 8m x 5m [Substitute values]
A = 40 \(m^2\) [Multiply]
Therefore the area of the entire section of wood is 40 \(m^2\) .
The area of a rectangle gives us the idea of space that it occupies in two dimensional space. Practically, if someone has to make a carpet, a door or any object that is rectangular in shape, they would need to find the measure of the surface area of that object.
Perimeter is the length of the boundary of a rectangle, and the area is the measure of space within that boundary.
When finding the area, the values of the length and width are being multiplied. Accordingly, even the units are multiplied. So, if the length and width are in feet for example, , the result is in square feet.
The area of a dice cannot be found using this formula as this formula is only applicable for two-dimensional figures. A dice is a three dimensional shape. However, the area of one face of the dice can be found using this formula.