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Q3:
If the circumference of a circular sheet is $154$ m, find its radius. Also find the area of the sheet. (Take $\pi = \frac{22}{7}$)

Solution :

Given Variables & Initial Setup

We are analyzing a two-dimensional circular sheet with the following given parameters:

  • Circumference ($C$) = $154$ m
  • Mathematical constant ($\pi$) $\approx \frac{22}{7}$

Our objective is to determine the radius ($r$) and the total enclosed area ($A$) of the circular sheet.

r = 24.5 m Circular Sheet Geometry Circumference (C) = 154 m Area (A) = 1886.5 m²

Step 1: Calculating the Radius of the Circular Sheet

The circumference of a circle is defined as the continuous linear distance forming the boundary of the closed geometric figure. [Per standard Euclidean geometry, the formula relating circumference to radius is $C = 2\pi r$].

Substituting the given values into the equation:

$154 = 2 \times \left(\frac{22}{7}\right) \times r$

Multiplying the constants on the right side of the equation:

$154 = \frac{44}{7} \times r$

To isolate the variable $r$, we multiply both sides of the equation by the reciprocal of $\frac{44}{7}$, which is $\frac{7}{44}$:

$r = 154 \times \frac{7}{44}$

We can simplify the fraction by recognizing that both $154$ and $44$ are divisible by $22$:

$r = \left(\frac{154}{22}\right) \times \left(\frac{7}{2}\right)$

$r = 7 \times \frac{7}{2}$

$r = \frac{49}{2} = 24.5 \text{ m}$

Step 2: Calculating the Area of the Circular Sheet

The area of a circle represents the total two-dimensional space enclosed within its circumference. [The fundamental formula for the area of a circle is $A = \pi r^2$].

Substituting $r = \frac{49}{2}$ m and $\pi = \frac{22}{7}$ into the area formula to maintain absolute precision before the final decimal conversion:

$A = \frac{22}{7} \times \left(\frac{49}{2}\right)^2$

Expanding the squared term:

$A = \frac{22}{7} \times \frac{49}{2} \times \frac{49}{2}$

Executing the arithmetic simplification by cross-canceling the terms in the numerator and denominator:

  • Divide $22$ by $2$ to get $11$.
  • Divide $49$ by $7$ to get $7$.

$A = 11 \times 7 \times \frac{49}{2}$

$A = 77 \times \frac{49}{2}$

$A = \frac{3773}{2}$

Converting the improper fraction to a precise decimal:

$A = 1886.5 \text{ m}^2$

Final Solution: The radius of the circular sheet is $24.5$ m, and its total enclosed area is $1886.5$ m$^2$.


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