Can this probability question be solved using $3$D cube volume? 
A dice is rolled $3$ times. Find probability of getting a larger number than the previous number each time.

The answer in the book is : $^6C_3/6^3=20/216$. 
But I am thinking of solving it using geometry. Had the question been of $2$ dice rolls, I would have drawn a $6\times6$ grid of possible outcomes and found area of upper right triangle. Since it is for $3$ trials, can we draw a $6\times6\times6$ cube and then find the volume to find the probability?

Yes. You're basically listing all possible $6^3$ outcomes and finding which ones result in "success". You can use the geometric structure to make the problem more intuitive, which you are "allowed" to do because all outcomes here are equiprobable, and each die roll is independent of the others.
