The Area of a Rectangle

A rectangle is a quadrilateral in which all four angles are right angles and the opposite sides are equal in length.

Let $ABCD$ be a rectangle having $AB = a$ and $BC = b$. Let $AC$ be the diagonal which divides the rectangle into two right triangles, $\Delta ABC$ and $\Delta ADC$. $\begin{gathered} {\text{Area of }}\Delta ABC = \frac{1}{2}AB \times BC = \frac{1}{2}a \times b \\ \Rightarrow {\text{Area of }}\Delta ADC = \frac{1}{2}DC \times AD = \frac{1}{2}a \times b \\ \end{gathered}$

$\begin{gathered} {\text{Area of rectangle }}ABCD = {\text{Area of }}\Delta ABC + {\text{Area of }}\Delta ADC \\ \Rightarrow {\text{Area of rectangle }}ABCD = \frac{1}{2}a \times b + \frac{1}{2}a \times b \\ \therefore {\text{Area of rectangle }}ABCD = a \times b = {\text{length}} \times {\text{breadth}} \\ \end{gathered}$

Example:

In exchange for a square plot of land, one of whose sides is $84$ m, a man wants to buy a rectangular plot $144$m long which is of the same area as the square plot. Find the width of the rectangular plot.

Solution:

One side of the square plot $= 84$ m.
The area of the square plot $= 84 \times 84 = 7056$ square m.
The length of the rectangular plot $= 144$ m.
The length of the rectangular plot $=$ area of the square plot.
The length of the rectangular plot $= 7056$ square m.

$\therefore$ the width of the rectangular plot $= \frac{{7056}}{{144}} = 49$ m

Example:

How many tiles $20$cm square will be required to pave a footpath $1$m wide going around the outside of a grassy plot $28$ m long and $18$ m wide.

Solution: The width of the foot path $= 1$ m.
The outside dimensions of the plot are $28 + 2 = 30$ m and $18 + 2 = 20$ m.

Now,
The outside area of the plot $PQRS = 30 \times 20 = 600$ square m.
The inside area of the plot $ABCD = 28 \times 18 = 504$ square m.
The area of the foot path $=$ outside area $–$inside area $= 600 – 504 = 96$ square m.
The area of one tile $= \frac{{20}}{{100}} \times \frac{2}{{100}} = 0.04$ square m.

The number of tiles required $= \frac{{96}}{{0.04}} = 2400$ tiles.