# The Area of a Square

A square is a quadrilateral whose four angles are all right angles and whose all sides are equal.

Let $ABCD$ be a square with each side in length equal to $a$ and $AC$ is a diagonal which divides the square $ABCD$ into two equal triangles, $\Delta ABC$ and$\Delta ACD$. Since $ABC$ is a right triangle, therefore:

$\begin{gathered} {\text{Area of }}\Delta ABC = \frac{1}{2}AB \times BC \\ \Rightarrow {\text{Area of }}\Delta ABC = \frac{1}{2}a \times a = \frac{1}{2}{a^2} \\ \end{gathered}$

Similarly,
$\begin{gathered} {\text{Area of }}\Delta ACD = \frac{1}{2}CD \times AD \\ \Rightarrow {\text{Area of }}\Delta ACD = \frac{1}{2}a \times a = \frac{1}{2}{a^2} \\ \end{gathered}$

$\begin{gathered} {\text{Area of the Square}} = {\text{Area of }}\Delta ABC + {\text{Area of }}\Delta ACD \\ \Rightarrow {\text{Area of the Square = }}\frac{{\text{1}}}{{\text{2}}}{a^2} + \frac{1}{2}{a^2} = {a^2} \\ \therefore {\text{Area of the Square = (one side}}{{\text{)}}^{\text{2}}} \\ \end{gathered}$

Example:

A chess board contains $64$ equal squares and the area of each square is $6.25$ square cm. A border around the board is $2$cm wide. Find the length of the side of the chess board.

Solution:

There are $64$ equal squares, each of area $6.25$ square cm.
The total area bounded by $64$ squares $= 64 \times 6.25 = 400$ square cm.
One side of the square containing small squares $= \sqrt {400} = 20$ cm.
The width of the border $= 2$ cm.

The length of the side of the chess board $= 20 + 2(2) = 24$ cm.