Question

The perimeter of the longest circle that can be inscribed in a rectangle of sides \(161 \, \text{cm}\) and \(296 \, \text{cm}\), in cm, is (take \( \pi \approx \frac{22}{7} \)).

• A. \(253 \)
• B. \(465\frac{1}{7} \)
• C. \(506 \)
• D. \(930\frac{2}{7} \)

Hint:

In geometry problems involving inscribed circles, always check which dimension restricts the circle — it is always the smaller side of the rectangle.

Answer 1

The Correct Option is C

Concept: The largest circle that can be inscribed inside a rectangle is the circle which touches all four sides of the rectangle. For such a circle, its diameter is always equal to the smaller side of the rectangle because the circle must fit completely inside the rectangle without crossing any boundary.

Step 1: Understanding the geometry of the inscribed circle.
In a rectangle, a circle can only be fully inscribed if it touches all four sides. This means the height and width both constrain the circle. Therefore, the maximum possible diameter of such a circle is limited by the smaller side of the rectangle.

Step 2: Identify the dimensions of the rectangle.
The given rectangle has: \[ \text{Length} = 296 \, \text{cm}, \quad \text{Breadth} = 161 \, \text{cm} \] Since the circle must fit inside both dimensions, the limiting factor is the smaller side: \[ \text{Diameter of circle} = \min(161, 296) = 161 \, \text{cm} \]

Step 3: Relating diameter and radius.
The radius of the circle is: \[ r = \frac{161}{2} = 80.5 \, \text{cm} \] However, for circumference calculation, we directly use diameter since: \[ C = \pi d \]

Step 4: Apply the formula for circumference of a circle.
The perimeter of a circle (circumference) is given by: \[ C = 2\pi r = \pi d \] Substituting values: \[ C = \frac{22}{7} \times 161 \]

Step 5: Simplify the expression carefully.
Break 161 as: \[ 161 = 7 \times 23 \] So, \[ C = \frac{22}{7} \times (7 \times 23) \] Cancel 7: \[ C = 22 \times 23 \] Now multiply: \[ C = 506 \] Final Answer: \(506\)