# Is the the distance squared inversely proportional for force of gravity?

My professor said that since mass is directly proportional to the volume, if we triple the radius we would consequently increase the mass by 27 times. Afterwards we would just plug the numbers newtons law of gravitation formula. The final part of the problem requires the gravitational force formula:

\begin{aligned} F&=\ \frac{GM_{1}M_{2}}{R^2} \\ \\ F_{R} &=\ \frac{G(M)(M)}{R^2} \\ \\ F_{3R} &=\ \frac{G(27M)(27M)}{(3R)^2} \\ \\ &=\ \frac{G(27M)(27M)}{9R^2} \\ \\ &=\ \frac{81G(M)(M)}{R^2} \\ \\ &=\ 81F_{R} \end{aligned}

​​ ​​ However I don't understand why he placed R instead of 2R for the second step. Shouldn't it be 2 R because R should be the distance from the center of mass? Same thing with the third step. Shouldn't R be 6R?

Yes, it should. This is a case where two wrongs made a right. Your instructor made the same mistake twice, and those mistakes happened to cancel. Use $\frac {GM^2}{(2R)^2}$ and $\frac{G (27M)^2}{(6R)^2}$ and you'll get the same ratio, an 81-fold increase in the gravitational force.