Question

There are two spheres. If the radius of second sphere is four times the radius of first spheres, then what will be the ratio of the surface area of second sphere to first sphere?

• A. $1:16$
• B. $4:1$
• C. $1:4$
• D. $16:1$

Answer 1

The correct option is D $16:1$

The formula used to calculate the surface area of sphere with radius $r$ is

$$S=4\pi r^2,$$
Let the surface area of first sphere be denoted as $S_1$ and radius as $r_1$,
therefore its surface area will be given as,
$$S_1=4\pi r_1^2$$

Similarly, assume that the surface area of second sphere be $S_2$ and its radius $r_2$, therefore the surface area of second sphere will be,

$$S_2=4\pi r_2^2$$It is given that the radius of second sphere is $4$ times the radius of first sphere, i.e., $$r_2=4r_1$$

Therefore, substitute $4r_1$ for $r_2$ in the formula of surface area of second sphere, we get
$$S_2=4\pi \left(\right(4r_1{)}^2)$$

The ratio of $S_2$ to $S_1$ will be commuted as follows,
$$\begin{array}{cc}\frac{S_2}{S_1}& =\frac{4\pi \left(\right(4r_1{)}^2)}{4\pi r_1^2}\\ & =\frac{64}{4}\\ & =\frac{16}{1}\end{array}$$
Hence, the ratio of surface area of second sphere to that of first sphere is $16:1$.