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I plotted $z = x^2+y^2$ and got this shape. Is there a name for this shape?

How about $z \ge x^2 + y^2$ ?

enter image description here

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The surface is a "circular paraboloid" (a special case of the "elliptic paraboloid"). Slices by planes perpendicular to the $z$ axis are (concentric) circles, while slices by planes parallel to the $z$ axis are (identical) parabolas.

It's an example of a surface of revolution, in this case obtained by revolving the basic parabola shape (think $y = x^2$) about its axis.

The filled counterpart would simply be a "solid circular paraboloid".

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A paraboloid, see the Wikipedia page: https://en.wikipedia.org/wiki/Paraboloid

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  • $\begingroup$ To be precise, an elliptic paraboloid, I think. $\endgroup$ – Old John Dec 1 '13 at 20:10

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