Question

which of the following statements about the area of a circle is true? choose the correct answer below. a. the area of a circle is always a whole number. b. the area of a circle is π times the radius squared. c. the area of a circle is always larger than the area of a square where the diameter of the circle is the same as the length of the side of the square. d. the area of a circle is π times the diameter squared.

Explanation:

• Option A: The area of a circle involves $\pi$ (an irrational number), so it's not always a whole number. For example, a circle with radius 1 has area $\pi\approx3.14$, not a whole number.
• Option B: The formula for the area of a circle is $A = \pi r^2$, which means the area is $\pi$ times the radius squared. This is a fundamental formula in geometry.
• Option C: If the diameter of the circle (and side of the square) is $d$, the circle's area is $\pi(\frac{d}{2})^2=\frac{\pi d^2}{4}\approx0.785d^2$, and the square's area is $d^2$. So the circle's area is smaller, not larger.
• Option D: The formula is in terms of radius squared, not diameter squared. The diameter is $2r$, so $\pi$ times diameter squared would be $\pi(2r)^2 = 4\pi r^2$, which is incorrect.

Answer:

B. The area of a circle is $\pi$ times the radius squared.