#### Answer

r= 2 in.
h= 3 in.

#### Work Step by Step

Lateral area = 2 $\times$ $\pi$ $\times$ r $\times$ h
12 $\pi$ = 2 $\times$ $\pi$ $\times$ r $\times$ (r+1)
6 = $r^{2}$ + r
$r^{2}$ + r - 6 = 0
r = $\frac{-b + \sqrt (b^{2} - 4 ac)}{2a}$
a= 1
b= 1
c=-6
r = 2 in.
h = 2 in + 1 in. = 3 in.