What a lot of cumbersome maths!
The easy way is to look up in an engineers' reference book the relationship between pitch circle diameter and the distance between vertices. This is typically used when one has to set out the vertices by co-ordinates. This is usually given in the books for many regular polygons. For an equilateral triangle, the PCD multiplied by 0.86603 gives the distance between vertices, to a good-enough approximation.
If we are to be treated to a mathematical tutorial extending over two pages, let us at least have an elegant solution. A little geometric insight shows that the PCD = 4/3 X Square root (3/4 X D^2), where D is the distance between vertices. No need for a calculator with trancendental functions!
This type of trivial and badly edited article does MEW no favours. The mathematical approach is clumsy and makes the subject intimidating for novices. For non-beginners, there is no need for articles teaching Grannie to suck eggs, either. (The triangle is equilateral, not isosceles and the plural of 'apex' is either apices or apexes, not apeces, as printed. I was pleased to see evidence of apparently improved proof-reading under the new editor, but standards seem to have slipped dramatically. In a publication with an international readership, it is important to maintain high standards of English usage, spelling and punctuation. Sadly, the correct use of the apostrophe seems to be a vanishing art…).
MEW seems to be becoming an an organ for entertainment, rather than authoritative information. It may be interesting some to learn how the editor mucked about with welding equipment, but an article telling us how to do it properly would be infinitely more useful. Please can we have no more articles of the "Wot I did on me 'olidays" type?