Speaker Coverage Patterns - From what is the % derived?

Common speaker coverage patterns include 50% (or edge-to-center) and 20% (Diagonal edge-to-edge), but what is it "20%" or "50%" of?

I understand that I'm doing math on what may be essentially a jargon issue, but still...

I'm developing my own calculators for speaker coverage patterns and have been trying to integrate the "50% coverage" and "20% coverage" benchmarks...  but what are these percentages meant to refer to?

They don't seem to refer to the ratio of overlapped area of a neighboring speaker to a single speaker's coverage area.  The math was kinda painful, but in the end wasn't close.   (In any case, a "50%" overlap pattern overlaps noticeably less than 50% of its neighbor... just by eyeballing.) 

My gut assumed the following: given a line through the centers of two equal and overlapping circles; the ratio of the line length falling within the overlap to the length of the circles' diameter."  This means the 50% works.  (The neighboring pattern overlaps by half - 50% - of the diameter of the circle.)  However, the 20% figure should actually then be about 29.3% by this math.  More precisely:  ((-1 * SQRT(2))+2)/2.

I wonder if perhaps this is supposed to represent the aggregate amount of overlapped area within the pattern?  I haven't tried calculating that, but I'm willing to try if that's the intention.  But starting and ending where?  Within a square/rectangle defined by the centers of the outermost speakers, as shown here? (Edit: I solved it below as well.)

Is this what speaker coverage figures pertain to?

Does anyone know where these terms originated or what math they purport to represent?

(Update:  In the above image,  25.56% of the square is covered by overlaps in speaker coverage.  Looks like this is not the solution to my query, as the pattern is 'looser' than the "20%" overlap pattern.) 

(Update: I ended up solving the overlapping area by percentage for several different permutations of overlapping circles and bounding squares.  I didn't come up with anything that would match the "50%" and "20%" patterns.  See Below.  My calculated percentages are in the lower right of each bounding box.)

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Go to the profile of Greg Bronson CTS-D
10 months ago

Hi Christopher - interesting question.  Per CTS-D Exam Guide 2nd Edition starting ~page 145, it's an approximation of speaker spacing (creating overlap) delta of the distance (not area) between two speakers from otherwise "edge to edge" coverage distance.  

Go to the profile of Christopher Eloranta
10 months ago

So, that suggests my gut was correct, and that "20%" spacing should really be called "29.3% spacing."