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.


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\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 144m 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 20cm square will be required to pave a footpath 1m wide going around the outside of a grassy plot 28 m long and 18 m wide.

 

Solution:


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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.