Sphere Surface Area Ratio Calculation
The surface area of a sphere is calculated using the formula $A = 4\pi r^2$, where $r$ is the radius.
If we have two spheres with radii $r_1$ and $r_2$, their surface areas $A_1$ and $A_2$ are:
- $A_1 = 4\pi r_1^2$
- $A_2 = 4\pi r_2^2$
Calculating the Ratio
We are given the ratio of the radii, $r_1 : r_2 = 2 : 3$.
To find the ratio of their surface areas ($A_1 : A_2$), we divide the two formulas:
$ \frac{A_1}{A_2} = \frac{4\pi r_1^2}{4\pi r_2^2} $The $4\pi$ terms cancel out, leaving:
$ \frac{A_1}{A_2} = \frac{r_1^2}{r_2^2} = \left(\frac{r_1}{r_2}\right)^2 $Final Ratio Determination
Substitute the given ratio of the radii ($r_1 : r_2 = 2 : 3$):
$ \frac{A_1}{A_2} = \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9} $Therefore, the ratio of their surface areas is 4 : 9.
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