A simple way is to minimize calculations and let Illustrator find how long parts are needed. The slices are parts of cone surfaces. You should understand it if you watch the following revolved polyline. It's a half of approximated sphere.
The orange stripe can be cut as an edge slice of a circle sector which forms the surface of a cone. Some calculations are needed. You must find the sector angle of the circle.
In the leftmost drawing you can find AC, BC and CD. AC can be got exactly by inserting a node to the crossing A and removing the extras with the direct selection tool.
AC was drawn originally by extending the polyline segment BC holding the shift key and dragging the corner towards A.
Document Info panel's subpanel "Objects" gives the following lengths:
AC=107,715mm BC=11,14mm and DC=41,22mm
The revolution path of point C is a circle with radius DC. That gives cone bottom perimeter = 258,99mm. (=multiplied by 2*pi) On the other hand that must be equal with the length of an arc which has radius = AC. The angle of the arc in radians =(258,99mm)/AC= 2,4044 radians = 137,76 degrees.
Direct formula for the sector angle in degrees is (DC/AC)*360. So here we have the orange stripe:
We have 2 circles. Radiuses: the red 107,715mm and the blue 11,14mm less (=BC). The sector angle is made by rotating a horizontal line to 137,76 degrees.
The rest of the needed stripes can be constructed in the same way. There's no reason why the stripes should have equal widths. It can be useful to make equator stripes and pole stripes 50% narrower and simplify them. The pole caps could be circle caps and the equator stripe could be one vertical cylinder instead of 2 opposite cones around the equator. Actually one gets a polygon just for it from Illustrator when he doubleclicks the polygon tool and inputs the values instead of dragging.
The example from 1970's seems to need a 36 gon as the starting point:
Actually you need the colored sides of it and the dashed square as a marker of the center. Red sides become curved stripes as the orange one was above. The green one is a round pole cap and the blue becomes a vertical cylinder (=a rectangle in a plane) symmetrically around the equator.
Automatization: Programming is needed. You may want to input the size of the sphere and and how many stripes it must have. I do not believe blending does the trick, altough I cannot prove that nobody will ever find good blending settings.
What to program: Trigonometric formulas for the stripe dimensions, drawing actual curves and the nesting (=placements for effective cutting). Programming and deriving the needed formulas are beyond the scope of this answer. But you can try this old-style Excel worksheet which contains your example case: https://www.dropbox.com/s/8dlssrkrdh7ebd9/spherestripes.xls?dl=0
It gives the dimensions of the needed stripes. Only the northern hemisphere and the equator are calculated, copy the stripes for the southern hemisphere.
Input the three starting parameters to yellow cells. Blue cells contain intermediate results.
If you happen to have a high end CAD program such as Inventor, Solidworks, SpaceClaim, Catia or other high cost stuff in the same league, you can revolve a N-gon to get a surface:
Only the northern hemisphere + the equator are revolved to keep the image simpler.
Because the surface stripes can be unfolded to plane, you at first split the surface to stripes. Assign some thickness to them and unfold them with the sheet metal functionality. Then you will have the stripes. Unfortunately I cannot show it in practice because I haven't such ($5000,-) software, the preceding images were only screenshots from entry level freeware CAD.
But nothing prevents to test the paperwork unfolder Pepakura that user Rafael suggested. Upvote him, if you see this useful:
I saved the revolved shape as STL file and imported it to Pepakura demo:
The circle in the left is the bottm cap. Not bad, I would say. Pepakura did the conversion in few seconds. It accepts also OBJ files.
In illustrator one can change the stokes, remove the unwanted parts, fill areas with the shape builder, duplicate those stripes which are needed for the southern hemisphere and move things to artistically acceptable places and positions. Here's what I got in a minute:
Filling with Shape Builder is actually a must, because the SVG from Pepakura contains a high number of unconnected line segments. There cannot be curves because STL file itself contains only straight lines.
