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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