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}


\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}


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.



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.