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Q1(c):
Find the circumference of the circles with the following radius: (Take $\pi = \frac{22}{7}$) (c) $21$ cm

Solution :

Given Variables & Initial Setup

We are tasked with determining the total boundary length, formally known as the circumference, of a circle. The following parameters are provided for the calculation:

  • Radius of the circle ($r$) = $21\text{ cm}$
  • Rational approximation for Pi ($\pi$) = $\frac{22}{7}$

Theoretical Foundation & Formula

The circumference ($C$) of a circle is directly proportional to its radius. The constant of proportionality relating the circumference to the diameter ($2r$) is $\pi$. [Per the fundamental axioms of Euclidean geometry, the ratio of a circle's circumference to its diameter is constant for all circles].

The governing formula for the circumference is:

$C = 2\pi r$

Geometric Representation

O r = 21 cm Circumference (C)

Step 1: Substitution of Values

Substitute the given values for $\pi$ and $r$ into the circumference formula:

$C = 2 \times \left(\frac{22}{7}\right) \times 21$

Step 2: Algebraic Simplification

To simplify the expression efficiently, we can divide the radius ($21$) by the denominator of $\pi$ ($7$). [By the associative and commutative properties of multiplication, we can group the terms to facilitate integer division].

$C = 2 \times 22 \times \left(\frac{21}{7}\right)$

$C = 2 \times 22 \times 3$

Step 3: Final Multiplication

Multiply the remaining integer values sequentially to find the total length:

$C = 44 \times 3$

$C = 132$

Because the radius is provided in centimeters (cm), the circumference—being a one-dimensional measure of length—must also be expressed in centimeters.

Final Solution: The circumference of the circle is $132\text{ cm}$.


More Questions from Class 9 Mathematics Coordinate Geometry EXERCISE 9.2


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