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The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the following statements is true?

  1. The circle has the largest area
  2. The square has the largest area
  3. The equilateral triangle has the largest area
  4. All the three shapes have the same area
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Migrated from GO Mechanical 4 years ago by Arjun

3 Answers

Best answer
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For a given perimeter circle has the largest area.

Similarly, for a given surface area, a cube has the largest volume.

So, correct answer: A
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5 votes

Let's take $100$ units as the perimeter

Circle:

  • $2πr = 100$
  • $\implies r = 100/(2π) = 100/(2\times 3.14) = 15.92$ units
  • Area $= πr^2$
  • $=3.14\times 15.92\times 15.92 = 795.82$ square units

Square:

  • $4a = 100$
  • $\implies a = 25 $units
  • Area $= a\times a = 25\times 25 = 625$ square units

Equilateral Triangle:

  • $3s = 100$
  • $s = 100/3 = 33.33$ units
  • Area $=\frac{\sqrt 3 }{ 4}  s^2$
  • $=\frac{\sqrt 3}{4} \times  33.33 \times 33.33$
  • $ = 481.029$ square units

$795.82 > 625 > 481.029$

Circle Area > Square Area > Equilateral Triangle Area

Hence, A is Correct.

edited by
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Let radius of circle = r
Perimeter of circle = 2*pi*r
Area of circle = pi*r*r

Edge of square = a
Perimeter of square = 4*a
Area of square = a*a

Edge of Triangle = x
Perimeter of Triangle = 3*x
Area of Triangle = root(3)x*x/4

 

Perimeter is same i.e. 2*pi*r = 4a = 3x =t
r = t/2*pi,
a = t/4
x = t/3

Area of circle, square, triangle are t*t/4*pi, t*t/16, root(3)t*t/36 = t*t (1/4*pi, 1/16, root(3)/36)

4*pi < 16 < root(3)/36
i.e. Area (circle) > Area (square) > Area (triangle)
Answer:

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