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:


square-01

\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 2cm 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.