Vertex O of the square OABC is situated at the center of the circle. If arc AC = 3π, then the perimeter of the square is

Appeared in: UPPSC-2021

Explanation

The central angle (∠AOC) subtended by the arc AC is 90°, which is the angle at a vertex of the square.
To use the arc length formula (L = rθ), the angle must be in radians. 90° is equivalent to π/2 radians.
Given the arc length L = 3π, we can set up the equation: 3π = r * (π/2).
Solving for the radius 'r' gives r = 6.
Since the vertex O is the center and A is on the circle, the side of the square (OA) is equal to the radius.
The perimeter of the square is 4 times the side length, so Perimeter = 4 * 6 = 24.

Option A: A perimeter of 18 would imply a side length of 4.5 (18 ÷ 4). This value does not result from the correct application of the arc length formula with the given values.
Option B: The value 247 is extremely large and does not logically follow from any calculation related to the problem's parameters. It is likely a typographical error in the options.
Option D: A perimeter of 32 would mean the side of the square is 8 (32 ÷ 4). This would be the result if the radius was calculated to be 8, which is incorrect. An error such as forgetting to divide by 2 in the angle (e.g., r = 3π / π * 2) could lead to a wrong radius.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Calculating the perimeter of a square using the arc length of a circle as background academic context rather than a clinical decision trigger.
This question tests foundational mathematical and logical reasoning skills.
While not directly clinical, proficiency in geometry and algebra is essential for nurses for tasks like accurate medication dosage calculations, interpreting graphical data on monitors, and understanding measurements for fluid balance.
What if the shape was a regular hexagon inscribed with one vertex at the center? The central angle would be 60° (π/3 radians). If the arc length between two vertices on the circle was 3π, the radius would be r = 3π / (π/3) = 9, and the perimeter would be 6 * 9 = 54.

How to Approach the Question

First, identify the geometric shapes and their relationship: a square with one vertex at the center of a circle.
Recognize that the sides of the square originating from the center (OA and OC) are also radii of the circle.
Determine the central angle subtended by the arc AC. Since it's a square, this angle (∠AOC) is 90°.
Recall the formula for arc length: L = rθ, where θ must be in radians.
Convert the 90° angle to radians: 90° * (π/180) = π/2.
Substitute the given arc length (3π) and the angle in radians (π/2) into the formula to solve for the radius (r).

Concept Tested & Keywords

Concept Tested: Calculating the perimeter of a square using the arc length of a circle.
Stem keywords: square, OABC, center of the circle, arc AC, perimeter
Lead-in keywords: then the perimeter of the square is

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