A sphere must be created in two halves in Origin. To do so, open two empty matrix windows (called Matrix1 and Matrix2). Then, perform the following operations:

1 - Select Matrix1 and then the Matrix => Set (Matrix) Values menu.
2 - Enter in a ternary operator similar to the following:

sqrt(x^2+Y^2)>10?(0/0):(+1*sqrt(100-X^2-Y^2))

3 - Click OK (or Do It depending on your version).
4 - Activate Matrix2 and perform steps 1 - 3 using an equation similar to the following:

sqrt(x^2+Y^2)>10?(0/0):(-1*sqrt(100-X^2-Y^2))

5 - Plot Matrix2 (which represents the bottom half of the sphere) first. To do that, activate Matrix2 and select the desired plot type from the Plot3D menu.
6 - Add Matrix1 to the graph through the Layer dialog box. To do that, double-click on the layer 1 icon in the top left corner of the graph. Then, select Matrix1 from the list of Available Datasets and click the right arrow (=>) button. This moves it into the Layer Contents. Click OK to complete the task.

Note: There are two additional steps which must be taken to make the sphere look correct.

First, the Z axis scale must be updated to accommodate both surface plots. In this example, the Z axis should be -10 to 10. To update it, select Format => Axes => Z Axis and update the From and To text boxes on the Scale tab.

Second, if Speed Mode is on, turn it off. To do that, select Format => Layer and disable the "Matrix data, maximum points per dimension" check box on the Size/Speed tab. If you are using an earlier version of Origin (i.e. prior to 6.0), Speed Mode will not affect your plot. You can therefore skip this step.