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Q1(b):
Find the circumference of the circles with the following radius: (Take $\pi = \frac{22}{7}$) (b) $28$ mm

Solution :

Given Variables & Initial Setup

We are tasked with determining the circumference of a circle given its radius. The fundamental parameters provided for this geometric system are:

  • Radius of the circle, $r = 28 \text{ mm}$
  • Approximation for Pi, $\pi = \frac{22}{7}$

Theoretical Foundation: Circumference of a Circle

The circumference ($C$) of a circle is defined as the continuous line forming the boundary of the closed geometric figure. [Per the geometric definition of a circle's perimeter in Euclidean space], the circumference is directly proportional to its radius, with the constant of proportionality being $2\pi$. The governing formula is:

$C = 2\pi r$

O r = 28 mm Circumference (C) = ?

Step 1: Substitution of Values

We substitute the given values of $r$ and $\pi$ into the standard circumference formula:

$C = 2 \times \left(\frac{22}{7}\right) \times 28$

Step 2: Algebraic Simplification

To optimize the calculation, we first analyze the terms for common factors. The radius ($28$) is a multiple of the denominator ($7$) in our fractional value of $\pi$. [By the fundamental properties of rational numbers], we can divide $28$ by $7$ to simplify the expression before proceeding with multiplication:

$\frac{28}{7} = 4$

Substituting this simplified integer back into the equation yields:

$C = 2 \times 22 \times 4$

Step 3: Final Calculation

We now perform the sequential multiplication of the remaining scalar quantities:

First, multiply the constants:

$2 \times 22 = 44$

Next, multiply the result by the simplified radius factor:

$C = 44 \times 4$

$C = 176$

Since the radius was provided in millimeters ($\text{mm}$), the circumference, being a one-dimensional measure of length, must carry the identical unit.

Final Solution: The circumference of the circle is $176 \text{ mm}$.


More Questions from Class 9 Mathematics Coordinate Geometry EXERCISE 9.2


CBSE Solutions for Class 9 Mathematics Coordinate Geometry


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