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Q1(a):
Find the circumference of the circles with the following radius: (Take $\pi = \frac{22}{7}$) (a) $14$ cm

Solution :

Given Variables & Initial Setup

We are tasked with determining the circumference of a circle based on its given radius. The parameters established for this geometric calculation are:

  • Radius of the circle, $r = 14 \text{ cm}$
  • Rational approximation for Pi, $\pi = \frac{22}{7}$

Geometric Visualization

The following diagram illustrates the circle with its center $O$, the given radius $r$, and the boundary representing the circumference $C$.

O r = 14 cm Circumference (C)

Step 1: Stating the Governing Formula

The circumference $C$ of a circle is defined as the total linear distance of its continuous boundary. [Per the fundamental theorems of Euclidean geometry], the circumference is directly proportional to its radius, mathematically expressed by the formula:

$C = 2\pi r$

Step 2: Substitution of Values

Substituting the given scalar values into the circumference formula yields:

$C = 2 \times \left(\frac{22}{7}\right) \times 14$

Step 3: Algebraic Simplification

To evaluate the expression efficiently, we first divide the radius by the denominator of the $\pi$ fraction [Applying the associative property of multiplication to isolate compatible factors]:

$C = 2 \times 22 \times \left(\frac{14}{7}\right)$

Since $14$ is a multiple of $7$, the fraction simplifies to an integer:

$C = 2 \times 22 \times 2$

Proceeding with the multiplication sequentially from left to right:

$C = 44 \times 2$

$C = 88$

Step 4: Dimensional Analysis

The radius is provided in centimeters ($\text{cm}$). Because the factor $2\pi$ is a dimensionless mathematical constant, the resulting calculated circumference strictly inherits the unit of the radius. Therefore, the final unit is centimeters ($\text{cm}$).

Final Solution: The circumference of the circle is $88 \text{ cm}$.


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