[amp_mcq option1=”their areas are equal” option2=”the area of the circle is greater than the area of the square” option3=”the area of the square is greater than the area of circle” option4=”the area of the circle is two times the area of the square” correct=”option1″]<\/p>\n
The correct answer is (a).<\/p>\n
The perimeter of a circle is $2\\pi r$, where $r$ is the radius of the circle. The perimeter of a square is $4s$, where $s$ is the side length of the square.<\/p>\n
If a circle and a square have the same perimeter, then $2\\pi r = 4s$. Solving for $r$, we get $r = \\frac{2s}{\\pi}$.<\/p>\n
The area of a circle is $\\pi r^2$. The area of a square is $s^2$.<\/p>\n
Substituting $r = \\frac{2s}{\\pi}$ into the formula for the area of a circle, we get $\\pi \\left( \\frac{2s}{\\pi} \\right)^2 = \\frac{4s^2}{\\pi}$.<\/p>\n
Substituting $r = \\frac{2s}{\\pi}$ into the formula for the area of a square, we get $s^2$.<\/p>\n
Therefore, the area of the circle is equal to the area of the square.<\/p>\n
Option (b) is incorrect because the area of a circle is always less than or equal to the area of a square with the same perimeter. This is because the area of a circle is proportional to the square of its radius, while the area of a square is proportional to the square of its side length. Since the radius of a circle is always less than or equal to the side length of a square with the same perimeter, the area of a circle is always less than or equal to the area of a square with the same perimeter.<\/p>\n
Option (c) is incorrect because the area of a square is always greater than or equal to the area of a circle with the same perimeter. This is because the area of a square is proportional to the square of its side length, while the area of a circle is proportional to the square of its radius. Since the side length of a square is always greater than or equal to the radius of a circle with the same perimeter, the area of a square is always greater than or equal to the area of a circle with the same perimeter.<\/p>\n
Option (d) is incorrect because the area of a circle is never two times the area of a square with the same perimeter. This is because the area of a circle is proportional to the square of its radius, while the area of a square is proportional to the square of its side length. Since the radius of a circle is always less than or equal to the side length of a square with the same perimeter, the area of a circle is always less than or equal to the area of a square with the same perimeter.<\/p>\n","protected":false},"excerpt":{"rendered":"
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