A builder has 100 m of fencing wire. He can use this wire to enclose a circular garden or a square garden. If he wants to maximiz…

Question

A builder has 100 m of fencing wire. He can use this wire to enclose a circular garden or a square garden. If he wants to maximize the enclosed area, what is the approximate ratio of the area of the largest possible circular garden to the area of the largest possible square garden, using the entire 100 m of wire?

1. A. 1 : 1
2. B. 1.27 : 1 (Correct answer)
3. C. 1.57 : 1
4. D. 2 : 1

Correct Answer

Option B — 1.27 : 1

Detailed Solution & Explanation

The correct answer is 1.27 : 1.

Key Points

• The same 100 m of wire becomes the perimeter of each figure.
• Circle: 2πr = 100, so r = 100/2π ≈ 15.92 m, and area = πr² ≈ 795.8 m².
• Square: 4a = 100, so a = 25 m, and area = 25² = 625 m².
• Ratio = 795.8 / 625 ≈ 1.27 : 1.
• The exact value is π/4 inverted — for a fixed perimeter, circle area : square area = 4/π ≈ 1.273, a constant worth remembering.
• This is why a circle encloses the greatest area for any given perimeter, which is the principle the question is really testing.

Additional Information

• For a fixed perimeter, the circle encloses the maximum possible area — the isoperimetric property, and the reason the ratio here exceeds 1.
• The constant is exact: circle area : square area = 4/π ≈ 1.273 for equal perimeters, independent of the actual length.
• Among regular polygons with the same perimeter, area rises with the number of sides, approaching the circle as a limit.

प्रश्न (हिन्दी में)

एक बिल्डर के पास 100 मीटर बाड़ लगाने वाला तार है। वह इस तार का उपयोग एक गोलाकार बगीचे या एक वर्गाकार बगीचे को घेरने के लिए कर सकता है। यदि वह घिरे हुए क्षेत्र को अधिकतम करना चाहता है, तो पूरे 100 मीटर तार का उपयोग करके, सबसे बड़े संभावित गोलाकार बगीचे के क्षेत्रफल और सबसे बड़े संभावित वर्गाकार बगीचे के क्षेत्रफल का अनुमानित अनुपात क्या है?

1. A. 1 : 1
2. B. 1.27 : 1
3. C. 1.57 : 1
4. D. 2 : 1