In geometry, a square is a regular quadrilateral that has four equal sides and four angles which are

\begin{array}{l} 90^{\circ} \end{array}

in measurement. Thus the sum of all the internal angles of a square is equal to

\begin{array}{l} 360^{\circ} \end{array}

Diagonal of a square-

A square can have two diagonals. Each of the diagonal can be formed by joining the diagonally opposite vertices of a square. The properties of diagonals are as follows-

Both the diagonals are congruent (same length).
Both the diagonals bisect each other, i.e. the point of joining of the two diagonals is the midpoint of both the diagonals.
A diagonal divides a square into two isosceles right-angled triangles.

Length of the Diagonal-

Consider a square ABCD having sides equal to ‘a’ cm.

Let AD and BC be the two diagonals of a square.

Diagonal of a Square Formula

It is clearly visible that a diagonal divides the square into two right triangle, i,e,

\begin{array}{l} △ A C D \end{array}

and

\begin{array}{l} △ B D C \end{array}

Let us take any triangle of the two for calculating the length of the diagonal.

\begin{array}{l} △ A C D \end{array}

, applying the Pythagoras Theorem to obtain the diagonal length, we have

\begin{array}{l} A D^{2} = A C^{2} + C D^{2} \end{array}

……………(i)

we know AC = CD = a, …………………….(ii)

Substituting the value of (ii) in (i), we have

\begin{array}{l} A D^{2} = a^{2} + a^{2} \end{array}

\begin{array}{l} A D^{2} = 2 a^{2} \end{array}

\begin{array}{l} \Rightarrow A D = \sqrt{2} a \end{array}

Thus the length of the diagonal of the given square is-

\begin{array}{l} A D = B C = \sqrt{2} a \end{array}

Example 1- Find the length of a diagonal of a square whose side is 4 cm.

Solution- Given a = 4 cm

Length of a diagonal =

\begin{array}{l} 4 \sqrt{2} c m \end{array}

Example 2- Find the length of a diagonal of a square, given its area to be 64

$\begin{array}{l} c m^{2} \end{array}$

Solution- Given Area = 64

\begin{array}{l} c m^{2} \end{array}

We know, Area =

\begin{array}{l} a^{2} \end{array}

\begin{array}{l} a^{2} = 64 \end{array}

\begin{array}{l} \Rightarrow a = 8 \end{array}

Now Length of a diagonal =

\begin{array}{l} a \sqrt{2} = 8 \sqrt{2} c m \end{array}

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