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Here's what I know so far: Until 1912, there was no precise definition of the American flag, with the layout of the stars and exact dimensions of the components not officially defined. This changed with executive order 1637, which specified exact dimensions. The exact dimensions appear to have been supplied by the navy.

Most of the dimensions they supply are fairly self-explanaotory. The canton is 7 stripes tall, and 2/5 of the width of the flag, for example. Each star has an equal sized margin around it. But there's no obvious reason behind the diameter of the stars. This page on wikipedia says that it's 1/(10 * phi) the height of the flag, but there's no source or justification given, and the math doesn't quite work out.

This page has some correspondence between the Joint Army and Navy board and Taft, but doesn't shed any light on the star diameter.

Why is the diameter of the stars defined how it is?

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  • It would have changed in 1959, and again in 1960.
    – Spencer
    Commented Dec 8, 2023 at 23:00
  • Yes, but the size of the stars did not change.
    – user1454156
    Commented Dec 9, 2023 at 0:15
  • If you read the sentence before that on on the WP page, it says there were 66 different designs in use prior to that executive order. So presumably there in fact was no one "design" there, beyond cramming the requisite number of stars into the blue field and making it not look terrible so your rendition of the flag sells.
    – T.E.D. ♦
    Commented Dec 9, 2023 at 5:14

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The current design of the flag is given in Executive Order 10834 which was issued by President Eisenhower in August of 1959. It defines the size of the stars as 4/5 the width of the stripes. I would assume this was chosen to give a pleasing overall effect to the flag. I doubt if you will find a formula for this dimension.

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