If I understand correctly, you are asking for the "diameter" of a
triangle with sides 72 units.  Right?

First, to be clear, the diameter here is the diameter of the circle that
encloses the triangle.

Consider this diagram:

Here's a one-paragraph introduction to trigonometry:

  The basic trigonometric functions are defined for a right triangle
  (a triangle with one angle 90°).  (In this diagram, we're going to
  work with the triangle ABC.)
  They are functions of one of the two other angles. (Either BAC or
  ABC in the diagram above, but we're going to concentrate on BAC.)
  The hypotenuse is the long side, the one opposite the 90° angle.
  (AB in the diagram.)
  The adjacent side is the side that connects to the angle that you're
  working with, that isn't the hypotenuse.  (For angle BAC, this is AC.)
  The opposite side is the side that *doesn't* connect to the angle
  that you're working with.  (For angle BAC, this is BC.)
  The sine of an angle is the length of the opposite side divided by
  the length of the hypotenuse.  (The sine of BAC is BC/AB.)
  The cosine is the adjacent side divided by the hypotenuse. (The
  cosine of BAC is AC/AB.)
  The tangent is opposite divided by adjacent. (The tangent of BAC is
  BC/AC.)
  The "arc" functions (arcsine, arccosine, arctangent) take the ratio
  as an argument, and return the angle.

Now we can do your math.

(Values below to three significant figures.)
You want the side BD to be 72 units.
BC is half of that, 36 units.
The angle BAD is one third of a circle, 120°.
The angle BAC is one sixth of a circle, 60°.

sin(BAC) = BC/AB
sin(BAC)AB = BC  (multiply both sides by AB)
AB = BC/sin(BAC)  (divide both sides by sin(BAC))
AB = 36 / sin(60)  (substitute in the actual values of BC and BAC)
AB = 36 / 0.866  (value of sin(60) from the sin() function)
AB = 41.6 = radius of the circle
241.6 = 83.2 = diameter of the circle

A few more hints that can help to sanity-check your results:

  Note that the opposite and adjacent sides are always smaller than
  the hypotenuse, so sine and cosine are between zero and one.
  (Actually, between -1 and +1 when you consider triangles in the
  other quadrants.)
  Also note that when the angle is small, the sine is small, and when
  the angle is large the sine is large.
  And the other way around for cosine.Yup> 

So, sanity-checking our result...

  We expect the radius AB to be larger than the half-side BC.
  We're dividing BC by a cosine, which is a number less than 1, so the
  result will be larger than BC.
  Check.

That kind of sanity checking won't catch all errors, but it'll catch
cases where you divided instead of multiplying, and things like that.

Note that this basic diagram and set of operations will tell you all
sorts of things about the dimensions of regular polygons.  For a polygon
with n sides, BAD is 360/n, BAC is half of that, and you can use the
same sine and cosine ratios to relate any of the dimensions to each other.

For instance, if you know the distance that you want to the center of a
side, AC in this diagram, and you want the "radius" that you would pass
to OpenSCAD...

BAC = 360/n/2
cos(BAC) = AC/AB
AB * cos(BAC) = AC
AB = AC / cos(BAC)

Thus for a regular polygon with n sides, where you know the length of
AC, the radius AB is AC/cos(360/n/2).

Again, sanity checking...

  We expect AB to be larger than AC.
  We're dividing AC by a cosine, so the result will be larger than AC.
  Check.

  The more sides we have, the closer AB and AC will be.
  The more sides we have, the smaller the angle BAC will be.
  Smaller angles yield larger cosines; as the angle approaches zero
  the cosine approaches 1.
  Thus when the number of sides is large, the cosine approaches 1.
  As the cosine approaches 1, AC/cos(BAC) will approach AC.
  Check.

UNITS NOTE:  OpenSCAD works in degrees.  Most other programming
languages work in radians, and some calculators default to radians.  If
you're doing this math using some other tool, make sure you convert
appropriately:  360° = 2*PI radians.  Most environments provide
conversion functions.

which is the answer obtained from the original suggestion once the
extraineous *'s were removed.

On 5/23/2023 2:06 PM, gene heskett wrote:

which is the answer obtained from the original suggestion once the
extraineous *'s were removed.

... which is a nice double check.

I did see that original response, but it was magic - it was the AI's
answer to that particular problem, with no explanation of why it was the
answer or how to answer related questions.  I thought I'd fill in that
missing part.  One unobvious connection is that sin(60) is sqrt(3)/2 - but
that doesn't tell you how to do similar math with squares, pentagons, et
cetera.

Also, I don't know whether ChatGPT will ever get basic math wrong, but it
does get facts about the real world wrong
https://en.wikipedia.org/wiki/Hallucination_(artificial_intelligence),
so I would double-check anything that it says.

The descriptions of the ChatGPT algorithm seem to say that it puts
together snippets that it decides "go together" based on the data that it
had been fed.  They imply that it doesn't actually understand anything.  It
yields amazing results, but has severe limitations.  It seems like it is in
a sense Dissociated Press
https://en.wikipedia.org/wiki/Dissociated_press on ultra-steroids.

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Also, I don't know whether ChatGPT will ever get basic math wrong, but
it does get facts about the real world wrong
https://en.wikipedia.org/wiki/Hallucination_(artificial_intelligence),
so I would double-check anything that it says.