GATE Mechanical 2018 Set 2 | GA | Question: 4

Question

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

• A. The circle has the largest area
• B. The square has the largest area
• C. The equilateral triangle has the largest area
• D. All the three shapes have the same area

Comments

circle?

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Area of circle > Area of Square > Area of Triangle

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https://math.stackexchange.com/questions/2030161/how-to-relate-areas-of-circle-square-rectangle-and-triangle-if-they-have-same

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Let $a$ is the permiter of circle, sqaure and equilateral trinagle

Assume, $\text{Area(Circle) = Area(Square) = Area(equilateral triangle)}$

$2 \pi r = a \Rightarrow \text{radius of circle } r = \frac{a}{2\pi}$

$\text{Side of square = } \frac{a}{4}$

$\text{Side of triangle = } \frac{a}{3}$

 

$\pi (\frac{a}{2\pi})^2 = (\frac{a}{4})^2 = \frac{\sqrt{3}}{4} (\frac{a}{3})^2$

$(\frac{1}{4\pi}) = (\frac{1}{16}) = \frac{\sqrt{3}}{4} (\frac{1}{9})$

$0.079 > 0.062 > 0.048$

$\text{Area(Circle) > Area(Square) > Area(equilateral triangle)}$

Answer 1

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

Comments

@Arjun sir,

For given surface area SPHERE has the largest volume, not CUBE..

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how to prove it analytically ?

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1. For a given perimeter, a circle has the largest area.
2. For a given area, a circle has the smallest perimeter.
3. For a given surface area, a sphere has the largest volume.
4. For a given volume, a sphere has the smallest surface area.

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@s_dr_13  analytical proof: @Digvijay Pandey's answer below.

Answer 2

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.

Comments

It is correct- but for a given perimeter circle has the largest area - it is in the definition of circle.

Answer 3

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)