How do I model a 12 sided shape?

I think there're many ways to model it (e.g. creating one lower segment, arraying it along a curve and using a bridge tool to connect it with circle). I'll show you my solution though :).

Add a cylinder with 12 vertices (change its number in a tool shelf- T). Add a loopcut (Ctrl+R) and place it near the bottom of the cylinder. Then scale the bottom edge a bit. Next select all the horizontal edge loops and subdivide them twice (W-->Subdivide). Select every fourth vertex of the middle edgeloop and place them lower (G,Z). Now select the middle edge loop and press Alt+Shift+S,1. Do the same with the upper one. Finally add the additional edge loops to make the subsurf modifier crease the edges nicely.

Add a subsurf modifier to the cylinder and smooth it. Add two vertical edge loops at the corner of the cylinder, next merge some vertices (Alt+M) and dissolve some edges (X-->Dissolve) as pictured below.

Do it for the other 3 corners. Delete other part of the cylinder and use a mirror modifier to create 3 halfs with the edge loops created.

Start with a plane and make some ring cuts so that you obtain this :

Use proportional editing with spherical falloff to round the top of the plane :

Add an array modifier with a count of 12. Add a simple deform modifier and set it to bend of 360°.

Apply these modifiers and remove all doubles.

The simple deform modifier has set the center of the shape "somewhere". To find this location duplicate the lower ring of the shape, extrude and scale 0. The center is found :

Now come back to the shape and keep only the front part, deleting all the other vertices. Like this :

We now want to flatten the bottom part. To do this, select the bottom part, make the vertex at the extremity active and set the active element as pivot point. Then scale Y to zero (assuming you have kept the good face of the shape which faces to Y):

Now to remake the whole circle :

Remember we have made a discus previously. Set the 3D cursor position at the discus center position :

Now come back to the shape, enter edit mode and select all:

Go to top view, and hit spin (from the toolshelf panel) and set the spin parameters to 12, dupli and 360° :

Select all and remove doubles.

Now set it a bit more "shappy".

Extrude Z the top and arrange the proportions :

Now the shading. If you set the shading to smooth, the effect will disappear.

So to handle that simply add an edge loop :

You may also want to add some bevel (based on angles and with a low angle value, like below) to enforce some edges :

I had a terrible work day, so to relax, now, I'll have my try at this...

1. create the basic cylinder A, very detailed (the main body)
2. create another simple 12 sided-cylinder B, centered exactly over the other
3. scale the cylinder B on Z axis (eg: 1/3rd), move it down along Z axis so that its top is a bit over the A's bottom, and the bottom is a bit below the B's bottom: it should be about vertically centered on A cylinder bottom face.
4. now the fun part: edit the cylinder B and delete top and bottom faces

5. in edit mode, select B top edge ring and scale it so that's bigger than A cylinder body
6. in edit mode, select B bottom edge ring and scale it so that's samller than A cylinder body

7. in edit mode, select all cylinder B in edit mode, and extrude scaling -Z (so, only on X and Y axis) to get a 12-sided solid "ring" like this:

8. then in object mode, select cylinder A and set boolean modifier with difference to cylinder B

here below is the overall concept: to me, it's very simple...

Little bit different method no array no boolean modifier

Make a circle of 12sides

Extrude in z axis

1 loopcut between top and bottom surface

select the center loop and subdivide once

select the alternative vertex as shown in gif

pull selected vertex little upward in z axis

Bevel the vertex (ctrl+shift+B) to make a round shape

Scale down the lower part

Flip the normals (ctrl+n)