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Perimeter of a Square

We know that, a square is a 4-sided polygon with all sides equal.
So, the perimeter of a square = sum of its all four sides l

= 4 × length of its side l l
Let the length of each side of the square be l units.
l
The perimeter of a square = 4 × l units
If the perimeter of the square is denoted by P.

Then the rule for the perimeter of the square will be P = 4 × l units
Example 4: The side of an equilateral triangle is shown by a. Express the perimeter of the
equilateral triangle using a.

Solution: Since, all three sides of an equilateral triangle are equal.
a a
Let a be the side of an equilateral triangle.
Since, the perimeter of an equilateral triangle = 3 × length of its side a

Thus, the perimeter of the equilateral triangle = 3a units
Example 5: The diameter of a circle is a line which joins two points on the circle and also passes
through the centre of the circle. (In the adjoining figure, AB is a diameter of the circle whose cente
is C. Express the diameter of the circle (d) in terms of its radius (r).

Solution: Given: C is the centre of the circle, CP is the radius denoted by r and AB is the diameter
denoted by d.
Since, the diameter is twice the length of the radius of the circle. A
So, diameter = 2 × radius = 2 × r = 2r C r P

Therefore, 2r is the diameter of the circle in terms of its radius. B
Rules From Arithmetic

As the variables are the representatives of unknown numbers, so variables are added, subtracted,
multiplied, and divided like numbers. In algebra, the mathematical relation between variables
and numbers holds the properties of addition, subtraction, multiplication, and division as the
numbers hold in arithmetic.

1. Commutative property: We know that two numbers hold the commutative property of
addition and multiplication.

As, 3 + 6 = 9 or 6 + 3 = 9
⇒ 3 + 6 = 6 + 3 = 9 (Commutative property of addition)

Also, 3 × 6 = 18 or 6 × 3 = 18
Clearly, 3 × 6 = 6 × 3 = 18 (Commutative property of multiplication)

In the same way, variables always hold the commutative property of addition and
multiplication.
Let x and y be two variables which represent two unknown quantities.

Therefore, x + y = y + x and x × y = y × x or xy = yx.

105 Introduction to Algebra