To find the unit rate of books per month that Liam read, you divide the total number of books read by the number of months in a year.
Liam read a total of 36 books in 12 months.
So, the calculation is:
\[ \text{Books per month} = \frac{\text{Total books}}{\text{Months}} = \frac{36 \text{ books}}{12 \text{ months}} = 3 \text{ books per month} \]
Therefore, the unit rate is 3 books per month.