 # 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 = $$a\sqrt{2} = 8\sqrt{2} cm$$