Slanted cylinder on a base

Hi all, this is my first time using the forum, many thanks for reading. I would like to ask how to create a slanted cylinder at an angle θ as shown in below diagram? The rectangle on the ground is the base while the cylinder itself is actually slanted. So far I had tried to :

1. emboss a circle to form a cylinder (but cannot tilt it)

2. rotation emboss of a rectangle tilted at an angle, with the sides either perpendicular to one another (so it makes a tilted cylinder, not what I wanted)

3. rotation emboss a parallelogram (which forms a pointy tip at one end and a concave end, still not what I want)

4. tried to make a proper cylinder and cut it to the shape below but don't know how,

Also, I am required to duplicate this a couple of time with linear patterns and my mentor at work actually told me to put holes into the base and then assemble the base and cylinder together. I was thinking more in the line to just "grow" the cylinders out of the base. Please advise and many thanks for your assistance.