Question

The length of the circumference of a circle equals the perimeter of a triangle of equal sides, and also the perimeter of a square. The areas covered by the circle, triangle, and square are $c$, $t$, and $s$, respectively. Then,

• A. $s>t>c$
• B. $c>t>s$
• C. $t>c>s$
• D. $s>c>t$

Hint:

For geometrical problems involving areas, equate the perimeters and use basic area formulas to compare the areas.

Answer 1

The Correct Option is A

Let the side of the triangle be $a$, the side of the square be $b$, and the radius of the circle be $r$. The perimeter of the triangle is $3a$, the perimeter of the square is $4b$, and the circumference of the circle is $2\pi r$. Since the perimeter of the circle equals the perimeter of the triangle and square, we can derive the following relationships between the areas: - The area of the circle is $c = \pi r^2$.
- The area of the triangle is $t = \frac{\sqrt{3}}{4} a^2$.
- The area of the square is $s = b^2$.
From these relationships, we find that $s>t>c$.