Question : A piece of wire 132 cm long is bent successively in the shapes of an equilateral triangle, a square, and a circle. The area will be largest in the shape of:
Option 1: Circle
Option 2: Equilateral triangle
Option 3: Square
Option 4: Equal in all the shapes
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Correct Answer: Circle
Solution : The length of the wire is the same in all three cases, so it becomes the perimeter of each shape. (1). For an equilateral triangle with side length $a$, the perimeter = $3a$ $a = \frac{132}{3} = 44$ cm The area of an equilateral triangle $ = \frac{\sqrt{3}}{4}a^2 = \frac{\sqrt{3}}{4}(44)^2= 838.32$ cm 2 (2). For a square with side length $a$, the perimeter = $4a$. So, $a = \frac{132}{4} = 33$ cm The area of a square $ = a^2 = (33)^2= 1089$ cm 2 (3). For a circle with radius $r$, the circumference = $2\pi r$ $2\pi r=132$ ⇒ $r = \frac{132}{2\pi} $ The area of a circle $ = \pi r^2 = \pi \left(\frac{132}{2\pi}\right)^2= 1386$ cm 2 The area will be largest in the shape of a circle. Hence, the correct answer is 'Circle'.
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