The Douglas Triangle, Prime Roots, and e

The Douglas Triangle is a special right triangle with sides 1 ∶ φ2 ∶ φ √3, where φ is the Golden Ratio. Joining two of these on the hypotenuse produces the Douglas Rectangle, which has interesting properties. One such property is a simple yet accurate approximation for e, which differs from e by 0.000000979... . Rewriting the formula gives an equivalent using the first three primes.