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You don't need to close a 3D model for area calculation, however, please note that if you close a hole on a surface your area calculation will be greater than if you left the holes in.
Volume automatically fills holes (at least the ones it's able to fill) and can take a bit of time if you have lots of holes to fill. You might want to do this manually for more control. A new unstructured mesh is created in addition to the mesh you are measuring the volume of.
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