I like the drawing in the spreadsheet - makes it clearer. I think Gordon was making the point that if the guy wanted to find the included angle using measurements (the OP's goal), the sheet did not give this number directly.

I would suggest that the formula above is not strictly correct. Suppose two balls, radii of r and R (small and large, respectively)

We put the small ball into our taper (making sure we have a way to get that ball out!) and using a depth gage measure the distance from a reference surface to the ball as l (small L).

Put the big ball in and measure depth L.

The difference between the small ball surface, adn the large ball surface, let's call "m", for measured difference. m = l-L

But this is not the difference between ball centers. That would be m - R + r. Call it d, for distance: d = m -R + r

But this is not the distance between the tangent point of the balls, nor is the taper diameter at the tangent points of the balls equal to r or R!

Draw a picture, and you realize that the distance along centerline is the length of the hypotenuse of a triangle with the base parallel to the taper, and that the short side of the triangle is equal to R-r. But this triangle has an angle that is exactly equal to the half-angle of the taper!

So half angle is then arcsin((R-r)/(m-R+r)), and included angle is 2*arcsin((R-r)/(m-R+r)).

Choose two balls, one that almost goes to the bottom and one tha is close to the diameter of the large end. If you strictly use depth gages, both balls should be small enough not to protrude from the hole. But the important thing is to be able to measure the difference between the surface of the small ball and the large ball. That's m. The formula above should give you the angle (or half-angle, depending upon whether you include the factor 2).

Clear?