Question

The plane of a small circle on a sphere of radius 10 in. is in 5 in. from the center. Find the radius of the small circle; also find the great circle distance from its pole to its circumference.

Answer 1

Let's draw a picture of the problem:

From the given picrture, we can make the following right triangle:

where r denotes the radius of the small circle. Then, by applying Pythagorean theorem, we have

$r^2+5^2=10^2$

which gives

$\begin{gathered} r^2+25=100 \\ r^2=75 \end{gathered}$

then, the radius of the small circle is given by

$\begin{gathered} r=\sqrt[]{75} \\ r=8.66\text{ in} \end{gathered}$

Therefore, the first answer is r=8.66 inches.

Now, let's find the great circle distance from the pole to its circunference. So, let's make a picture of the problem:

where d is the distance from the pole to the greate circle. So, we have another right triangle:

So, by applying Pythagoren theorem, we have

$d^2=5^2+10^2$

which gives

$\begin{gathered} d^2=25+100 \\ d^2=125 \\ \text{then} \\ d=\sqrt[]{125} \\ d=11.18\text{ in} \end{gathered}$

Therefore, the answer for the second question is d = 11.18 inches