I just want to know the formula for the following question: If the dose rate from a sample of Ra-223 was found to be 20 mSv per hour at 10 micrometres, calculate the dose rate at 30 micrometres. Show your calculations. Give your answer to one decimal place.

I think it is 20 mSv x 10 micrometres / 30 micrometres, but I am not sure if I have to convert to metres or not (if not I get 6.6).

The point is that the radiation spreads out in a sphere. The dose rate at distance r is inversely proportional to the surface area of a sphere at distance r. Since the surface area itself is proportional to r^2, the dose rate is proportional to 1/r^2. three times as far, the dose rate is 1/9 as much, not 1/3.