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Can anyone help me create a star extrusion to circle?

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    $\begingroup$ I had to see the answers to understand what this was asking. Perhaps "star extrusion to circle " should be reworded as "connect the star vertices to the circle". $\endgroup$ – person27 Sep 18 '17 at 3:27
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Take the star, go to edit mode (Tab), select all vertices (A) and extrude it (E).

Stay in edit mode and keep the selection. Choose "Transform: To Sphere" (Shilft+Alt+S) and drag the mouse until you have a nice circle. Scale the circle (S) if you think it's too big.

Animation of the process

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First of all, your Circle should have the same amount of vertices as your Star (or it can have more but then it will be more complicated).

Also you can add Circle in Object Mode and join (Ctrl+J) it with Star or you can just add Circle while being in Star Edit Mode.

Then select both shapes (outer edges) and press CtrlE > Bridge Edge Loops.

After that you will want to select all and Recalculate Normals - CtrlN.

enter image description here

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    $\begingroup$ @ThomasWeller I'm using ScreenToGif app: screentogif.com $\endgroup$ – cgslav Sep 18 '17 at 11:36
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Enable the Looptools Addon.

enter image description here

Extrude the star and then use the loop tools>circle to turn the selected vertices into a circle.

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If you want this as Node Setup and using Sverchok Add-On in Blender:

https://gist.github.com/2bbbe7e82730a625ec876948eaea6620

node tree setup using sverchok to loft circle to star shape

star to circle extrusion

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Just for the sake of adding more answers...

You can also do this backwards. Start with a cylinder, select one of the edge loops, do checker deselect and scale the selected vertices to form the star shape.

enter image description here

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