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Since a UV sphere has poles, the best way to make a "perfect" sphere is subdividing a cube and casting it to a sphere, so there is no pinching. You can make a hemisphere by deleting the bottom half.

However, the vertices don't go around it in flat rings, so I can't use this method to make, say, a top third of a sphere.

Is there a better way to make a third of a sphere without a pole at the top?

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You can instead just use the boolean modifier in order to cut away the quad sphere. It should still keep all-quad geometry but the faces near the cut could be squished.

Important notes:

  • Using subdivision surface on a cube will not give you a perfect sphere. The corners will always be slightly lopsided.
  • Instead: Enable the "Extra Objects- Mesh" addon inside Edit->Preferences->Addons. This comes with Blender. Then, when you add a new object you can choose "Round Cube" inside the mesh section. Choose that, and switch the operator preset to "quadsphere" to make a truly round quadsphere.
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    $\begingroup$ I used a cast modifier after the subsurf, which gives a perfect sphere. I've stopped using subsurf in my actual models, instead just using higher vert counts. It makes modelling with exact measurements a lot easier. I ended up doing this, using a boolean. I was worried edge sliding the verts which effected the edge bevel would make it not smooth, but it didn't affect it at all. $\endgroup$ – JackHainsworth Sep 9 '20 at 17:09

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