Step 1: Understanding the Concept:
A circle is a fundamental geometric shape defined as the set of all points in a plane that are at a given distance (the radius) from a fixed point (the center).
The area of a circle measures the amount of two-dimensional space contained within its boundary, while the circumference is the total linear length of that boundary.
Both properties are intrinsically linked by the radius (\(r\)) and the mathematical constant $\pi$ (Pi), which represents the ratio of a circle's circumference to its diameter.
In this problem, we are provided with the area and must navigate through the radius to arrive at the circumference. This requires a two-step algebraic approach: first solving for \(r\) using the area formula and then using that \(r\) to compute the circumference.
Understanding this relationship is vital for mensuration problems, where dimensions are derived rather than directly provided.
Key Formula or Approach:
1. Area of a Circle (\(A\)) = \(\pi r^2\)
2. Circumference of a Circle (\(C\)) = \(2\pi r\)
3. Given Value: \(A = 154\) cm$^2$ and \(\pi \approx \frac{22}{7}\).
Step 2: Detailed Explanation:
We start by substituting the known area into the area formula to isolate the radius:
\[ 154 = \frac{22}{7} \times r^2 \]
To solve for \(r^2\), we multiply both sides of the equation by the reciprocal of \(\frac{22}{7}\), which is \(\frac{7}{22}\):
\[ r^2 = 154 \times \frac{7}{22} \]
We can simplify this by dividing 154 by 22. Since \(22 \times 7 = 154\), the expression becomes:
\[ r^2 = 7 \times 7 \]
\[ r^2 = 49 \]
Taking the square root of both sides to find the linear radius:
\[ r = \sqrt{49} = 7 \text{ cm} \]
Now that we have established the radius is 7 cm, we can calculate the circumference using the second formula:
\[ C = 2 \times \pi \times r \]
\[ C = 2 \times \frac{22}{7} \times 7 \]
In this expression, the 7 in the numerator and the 7 in the denominator cancel each other out perfectly:
\[ C = 2 \times 22 = 44 \text{ cm} \]
This logic holds that for any circle where the radius is a multiple of 7, the calculations involve integers, making it a favorite for exam setters.
Step 3: Final Answer:
The circumference of the circle is 44 cm. Therefore, the correct choice is Option (B).