To add to Ton and Jacob's (Hi, Ton and Jacob!) excellent methods, I'd like to share the one I use not only for segmented spheres but for fluted columns and for less regular divisions of any shape for which a cross-section of a 3D shape needs to be translated to a 2D view.
1) Draw a polygon with the appropriate number of points on the equator of the visible hemisphere of the final shape and a coincident circle. If you want the view of the final shape to be oblique; that is, not perpendicular to a face and not straight at a divider between segments, rotate the polygon the desired amount.
2) With Smart Guides and Snap to Point turned on, drag guides from the rulers to snap to the left and right sides of the circle and to each point on the lower (or upper: whichever defines the final visible segments) half of the polygon. Lock the guides and hide or delete polygon.
3) Direct Select (white arrow) the right (or left) anchor point of the circle and press delete to form a semi-circle.
4) Activate the Selection Tool (black arrow). Select the semicircle, copy, and paste in front, then drag the middle bounding box handle until the middle of the arc snaps to the next guide you come to. Paste in front and drag to the next guide until you run out of guides.
5) Hide or delete the guides, then use the Live Paint Bucket to color the segments and turn the segments into closed paths.
This is not quite instantaneous, but it's lots faster than explaining it, and I find it versatile.
Peter
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