Which is perhaps calculable, but if you tried to point Wolfram-Alpha at figuring the threshold of detection from an actor who is within a handful of seconds from having been exposed to full stage lighting...well, I think it would spit up.
In any case, a 20ma LED, even run down at around 6-7ma, is plenty bright enough. 20 degrees is a bit too narrow for the expected off-axis viewing and 40-60 is better.
Also, even more importantly, I "wasted" several hours drawing up plans for laser-cut acrylic when it turns out there are battery boxes available for about a buck fifty each. "Wasted," because I was at rehearsal anyhow and getting paid for that.
In any case, the project is spec'd, tested, and documented.
Working within parts available at the local Radio Shack (which, despite boarded-up windows appears to still be in business) the unit cost is about $2.50. Purchasing in bulk through DigiKey, I can get it down to $1.75 each. And build time is minimal.
As with most of my work, this is open hardware. The full Instructable is up, of course; at