# Diagonal Of A Square Formula

In geometry, a square is a regular quadrilateral that has four equal sides and four angles which are $90^{\circ}$ in measurement. Thus the sum of all the internal angles of a square is equal to $360^{\circ}$.

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.

It is clearly visible that a diagonal divides the square into two right triangle, i,e, $\bigtriangleup ACD$ and $\bigtriangleup BDC$.

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

In $\bigtriangleup ACD$, applying the Pythagoras Theorem to obtain the diagonal length, we have

$AD^{2} = AC^{2} + CD^{2}$ ……………(i)

we know AC = CD = a, …………………….(ii)

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

$AD^{2} = a^{2} + a^{2}$ $AD^{2} = 2a^{2}$ $\Rightarrow AD = \sqrt{2}a$

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

$AD = BC = \sqrt{2}a$

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 = $4 \sqrt{2} cm Example 2- Find the length of a diagonal of a square, given its area to be 64 \(cm^{2}$.

Solution- Given Area = 64 $cm^{2}$

We know, Area = $a^{2}$ $a^{2} = 64$ $\Rightarrow a = 8$

Now Length of a diagonal = $a\sqrt{2} = 8\sqrt{2} cm$