What is the value, in steradians, of the sum of the solid internal
angles of a random tetrahedron (as an analodue to the sum of the
internal angles of a random triangle being 2*pi radians)? Or is that
sum not constant? I'm pricing this at just 10 dollars but I will give
a handsome tip if you provide quite a good reply. Maybe you might be
interested in the question yourself.

js

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Clarification of Question by jeffsmith-ga on 12 Aug 2005 07:18 PDT

Sorry, the sum of angles of a triangle is pi radians (180 degrees). Oops!

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Request for Question Clarification by pafalafa-ga on 12 Aug 2005 16:34 PDT

Whew, Jeff.  It's been quite a number of years since my last attempt
at any solid geometry.  And I've never been all that handy with
steradians!

That said, it seems to me a tetrahedron is simply four triangles, so
that the sum of their angles would 4 times the sum of a single
triangle, or 4*pi radians.

That should hold for all tetrahedrons, regular or not.

Am I missing something?  Mathtalk-ga, where are you?  Any comments
would be appreciated.


pafalafa-ga

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Request for Question Clarification by pafalafa-ga on 12 Aug 2005 17:16 PDT

Ooops, just noticed the comments below, which apparently reflect quite
a bit more sophisitcated understanding of things than my own.

Hope you have all the information you need at this point.


Cheers,

paf

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Clarification of Question by jeffsmith-ga on 12 Aug 2005 17:55 PDT

Never mind, pafalafa. You showed your ability by providing this
incredible answer to my previous question. And I suppose I owe you 10
extra dollars for that, too, as pinkfreud had the kindness to answer
what seemed to be an offhand question about Hans Eysenck and received
the gratuity of 50 dollars for both his replies. This might have been
unfair to you, given the different qualities of your respective
replies to the wildcard characters question.

js

http://answers.google.com/answers/threadview?id=160247

12*arccos(1/3)

That is for a regular tetrahedron.  The sum is different for different
tetrahedrons.  You can see this by taking the limit of a very flat
tetrahedron.  One vertex approaches a solid angle of 2pi, the other 3
approach zero, so the sum of the solid angles tends toward 2pi, which
is different from the value for a regular tetrahedron.

Thank you for answering. This pretty much covers it. I guess it is
worth something, too, so expect a 2 dollar question from me.

js

Jeff,

Racecar isn't a Google Answers Researcher, so his assistance isn't
compensable. Looks as if you've gotten a nice freebie from one of the
site's most helpful users!

~Pink

Sorry about racecar. Having trouble with my eyesight lately. And I
thought I would age gracefully!

Hi jeffsmith-ga,

Glad my comment was helpful.  As you can see, I was interested in the
same question a while back.

Hi racecar. I take this question, when posed, to be one of the true
marks of an inquiring mind (especially if the questioner has not
studied Mathematics). Sorry again for mixing you up with a Google
researcher.

js (Dimitri B.)

Since pafalafa has apparently disdained to give just any sort of
answer to this question and collect 10 dollars, I will presently
cancel this question.

js (Dimitri B.)

Hi Jeff (or is it Dimitri?).

First off, I never properly thanked you for steering this question to
my attention.  So....Thanks!

I hope you don't feel I'm ignoring you.  But I certainly don't feel I
can post an answer to a question that...well...I haven't answered (and
don't really feel competent to answer).

So, I've just left the question here, content that you seem to have
gotten the answer you need from the comments section.

Hope to see you again one of these days.


paf

Ok, paf, I suppose that's clear enough. Thank YOU for taking the time
to post yet another comment. I will now retire the question.

D.B.