The shape of the galaxy is a circular, with diameter 100, 000 lights year,
so the radius is 50, 000 lights year, thus
Area of the galaxy is [tex]A(galaxy)= \pi r^{2}= \pi (50,000)^{2}=25*10^{8} \pi [/tex] (lights year square)
The area per civilisation is [tex] \frac{25*10^{8} \pi }{10,000}=25*10^{4} \pi[/tex]
from this result we can assume as a circle of Radius=[tex]25*10^{2}=2500[/tex] lights year.
and assume that each civilisation lives in a planet at the center of these smaller circles.
Check the figure attached.
Planet A, and B are closest in distance. They are each at the center of a circle with radius 2500, and they are tangent. So the distance of A to B is 2500+2500=5000 (light years).
Answer: 5000 light years
