Here we will discuss about the perimeter and area of a square and some of its geometrical properties.

Perimeter of a square (P) = 4 × side = 4a

Area of a square (A) = (side)^{2} = a^{2}

Diagonal of a square (d) = \(\sqrt{(\textrm{side})^{2}+(\textrm{side})^{2}}\)

= \(\sqrt{\textrm{a}^{2}+\textrm{a}^{2}}\)

= √2a

Side of a square (a) = √A = \(\frac{P}{4}\)

Some geometrical properties of a square

In the square PQRS,

PQ = QR = RS = SP

PR = QS

∠PQR = ∠QRS = ∠RSP = ∠SPQ = 90°.

PR and QS are perpendicular bisectors of each other.

Area of the ∆POQ = Area of the ∆QOR = Area of the ∆ROS = Area of the ∆SOP

Solved Examples on Perimeter and Area of a Square:

**1.** The perimeter and the area of a square are x cm and x cm\(^{2}\)
respectively.

(i) Find the perimeter.

(ii) Find the area.

(iii) Find the length of a diagonal of the square.

**Solution:**

Let a cm be the measure of a side of the square.

Then the perimeter = 4 a cm, area = a\(^{2}\) cm\(^{2}\)

From the question,

4a = x = a\(^{2}\)

or, a\(^{2}\) - 4a = 0

or, a(a - 4) = 0

Therefore, a = 0

or, a = 4

But, the side of a square ≠ 0

Hence, the side of the square = 4 cm

(i) Perimeter of a square = 4a

= 4 × 4 cm

= 16 cm

(ii) Area of a square = a\(^{2}\) cm\(^{2}\)

= 4\(^{2}\) cm\(^{2}\)

= 16 cm\(^{2}\)

(iii) Length of a diagonal = √2a

= √2 ∙ 4 cm

= 4√2 cm

= 4 × 1.41 cm

= 5.64 cm

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