A 12 m long wire is cut into two parts. One part is bent into a circle and the other part is bent into a square that just encloses the circle (the circle is inscribed in the square). What is the radius of the circle, expressed in metres, in terms of π?

Introduction / Context:
This geometry problem combines the concepts of perimeter and circumference with the idea of an inscribed circle. A single wire is used to form both a circle and a square, and the circle just fits inside the square. We are asked to determine the radius of the circle in terms of π. Such questions test spatial reasoning and the ability to relate different geometric shapes formed from the same total length.

Given Data / Assumptions:

• Total length of the wire is 12 metres.
• One part of the wire forms a circle of radius r metres.
• The other part forms a square that exactly encloses the circle (the circle is inscribed).
• The circle is tangent to all four sides of the square, so the diameter of the circle equals the side length of the square.
• We assume no wastage in bending the wire.

Concept / Approach:
When a circle is inscribed in a square, the diameter of the circle is equal to the side length of the square. If the radius is r, the diameter is 2r and the square has side 2r. The circumference of the circle uses part of the wire and equals 2πr. The perimeter of the square uses the rest of the wire and equals 4 times its side, that is 8r. The total wire length is therefore the sum 2πr + 8r, which must equal 12 metres. Solving this linear equation in r gives the required radius.

Step-by-Step Solution:
Step 1: Let r be the radius of the circle in metres.Step 2: The circumference of the circle formed by part of the wire is 2πr.Step 3: Since the circle is inscribed in the square, the side length of the square is equal to the diameter of the circle, which is 2r.Step 4: The perimeter of the square formed by the remaining wire is 4 × 2r = 8r.Step 5: The total length of the wire is 12 m, so 2πr + 8r = 12.Step 6: Factor out r: r(2π + 8) = 12.Step 7: Solve for r: r = 12/(2π + 8) = 12/[2(π + 4)].Step 8: Simplify the fraction: r = 6/(π + 4).

Verification / Alternative check:
To sense check the result, note that π is approximately 3.14, so π + 4 is roughly 7.14. Then r ≈ 6/7.14, which is a little less than 1 metre. This is reasonable, since the total wire length is 12 metres and both the circle and square must share that length. If r were much larger than 1, the combined perimeters would exceed 12 metres, which would be impossible. The expression 6/(π + 4) is therefore both algebraically correct and numerically sensible.

Why Other Options Are Wrong:
The option 12/(π + 4) corresponds to the side of the square or to twice the radius, not the radius itself. The option 3/(π + 4) is exactly half of the correct radius and results from mistakenly dividing by 4 instead of 2 at the final step. The expression 6/(π + 2√2) arises from confusing the geometry of a circle inside a square with the circle inside a different polygon. The last option 6/(π + 6) comes from an incorrect perimeter equation. Only 6/(π + 4) matches the correct derivation.

Common Pitfalls:
A typical error is to assume that the circumference of the circle equals the perimeter of the square or that they share equal wire lengths, which the question does not state. Another frequent mistake is misidentifying the relationship between the circle and square; for an inscribed circle, the diameter equals the side of the square, not the diagonal. Confusing this leads to wrong expressions in r. Carefully distinguishing diameter, side, and diagonal avoids these issues.

Final Answer:
The radius of the circle formed from the wire is 6/(π + 4) metres.