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.

Diagonal of a Square Formula

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 = \(a\sqrt{2} = 8\sqrt{2} cm\)

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