Question

the diameter of circle b is 3 times the diameter of circle a. complete the paragraph to compare the two circles. the circumference of circle b is times the circumference of circle a. the area of circle b is times the area of circle a.

Explanation:

Step1: Recall circle - related formulas

The circumference formula of a circle is $C = \pi d$ (where $d$ is the diameter), and the area formula is $A=\pi(\frac{d}{2})^2=\frac{\pi d^{2}}{4}$. Let the diameter of circle $A$ be $d_A$ and the diameter of circle $B$ be $d_B$. Given that $d_B = 3d_A$.

Step2: Compare the circumferences

The circumference of circle $A$, $C_A=\pi d_A$. The circumference of circle $B$, $C_B=\pi d_B$. Substitute $d_B = 3d_A$ into the formula for $C_B$, we get $C_B=\pi(3d_A)=3(\pi d_A)=3C_A$.

Step3: Compare the areas

The area of circle $A$, $A_A=\frac{\pi d_A^{2}}{4}$. The area of circle $B$, $A_B=\frac{\pi d_B^{2}}{4}$. Substitute $d_B = 3d_A$ into the formula for $A_B$, we have $A_B=\frac{\pi(3d_A)^{2}}{4}=\frac{\pi\times9d_A^{2}}{4}=9\times\frac{\pi d_A^{2}}{4}=9A_A$.

Answer:

The circumference of circle $B$ is 3 times the circumference of circle $A$. The area of circle $B$ is 9 times the area of circle $A$.