Question

1. what effect would each of the following have on the area and circumference of a circle? (a) doubling the diameter (b) halving the diameter (c) doubling the radius

Explanation:

Step1: Recall circle formulas

The area formula of a circle is $A = \pi r^{2}$ and the circumference formula is $C=2\pi r$, where $r$ is the radius and $d = 2r$ is the diameter.

Step2: Analyze case (a) - doubling the diameter

If the original diameter is $d$ and the new diameter is $d_{new}=2d$. Since $d = 2r$, if $d$ doubles, the new radius $r_{new}=2r$.
For the area, the original $A=\pi r^{2}$, the new $A_{new}=\pi(2r)^{2}=4\pi r^{2}$, so the area is multiplied by 4.
For the circumference, the original $C = 2\pi r$, the new $C_{new}=2\pi(2r)=4\pi r$, so the circumference is multiplied by 2.

Step3: Analyze case (b) - halving the diameter

If the original diameter is $d$ and the new diameter is $d_{new}=\frac{d}{2}$. Since $d = 2r$, the new radius $r_{new}=\frac{r}{2}$.
For the area, the original $A=\pi r^{2}$, the new $A_{new}=\pi(\frac{r}{2})^{2}=\frac{1}{4}\pi r^{2}$, so the area is multiplied by $\frac{1}{4}$.
For the circumference, the original $C = 2\pi r$, the new $C_{new}=2\pi(\frac{r}{2})=\pi r$, so the circumference is multiplied by $\frac{1}{2}$.

Step4: Analyze case (c) - doubling the radius

If the original radius is $r$ and the new radius is $r_{new}=2r$.
For the area, the original $A=\pi r^{2}$, the new $A_{new}=\pi(2r)^{2}=4\pi r^{2}$, so the area is multiplied by 4.
For the circumference, the original $C = 2\pi r$, the new $C_{new}=2\pi(2r)=4\pi r$, so the circumference is multiplied by 2.

Answer:

(a) Area is multiplied by 4, circumference is multiplied by 2.
(b) Area is multiplied by $\frac{1}{4}$, circumference is multiplied by $\frac{1}{2}$.
(c) Area is multiplied by 4, circumference is multiplied by 2.