After receiving a "brain," the Scarecrow in The Wizard of Oz famously says the following:

“The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side!”

Which as generations of know-it-alls have noted, is incorrect. That's a right triangle, you idiot!

Over the years I have a number of conflicting explanations for the mistake, all treated as the real reason. Among them:

  • this was intentional, and is intended to show the scarecrow is in fact as dim as always
  • this was intentional, because the filmmakers thought the word isosceles sounded "smarter"
  • this was mistake on the part of the actor, and the script said "right triangle."
  • this was a mistake on the part of the filmmakers

Is there any firm evidence or statements from those involved about which explanation is correct?

  • 57
    It's also sum of squares, not square roots. Sep 27, 2022 at 1:24
  • 27
    And it's not "any two sides", it's the two legs (sides adjacent to the right angle).
    – nanoman
    Sep 27, 2022 at 8:32
  • 17
    If it's not a right triangle, it's a wrong triangle. Sep 27, 2022 at 13:44
  • 3
    @PaulD. Waite no no no. It can also be a left triangle! Sep 28, 2022 at 21:57
  • 1
    Next: why does Morty get the first law of thermodynamics wrong in the first episode of Rick and Morty? Sep 29, 2022 at 6:18

3 Answers 3


There is a version of the script that had more to say to make it obvious that it was intentional, according to:

Hollywood Science: The Next Generation, From Spaceships to Microchips

(ISBN 331954215X, 9783319542157; Kevin R. Grazier, Stephen Cass; Springer, 2017)

So I went back to the scripts I’ve got here, and we can specially credit Noel Langley with that part of the script. I’ve got a draft dated April 18, 1938 and these are changes to a script he already did.

The sum of the square roots of any 
two sides of an isosceles triangle
 is equal to the square root of the
 remaining side: H-2-O plus H-2-S-
O-4 equals H-2-S-O-3 using pi-r
 squared as a common denominator. 
Oh joy! Oh rapture! What a brain!

—perhaps the blend of chemistry with geometry in that manner was too obvious. That line does, however, remove much of the doubt that the Scarecrow’s faulty exposition on the nature of the isosceles triangle is intentional on the part of the writers, and it also makes it clear that, in essence, the Wizard did very little.

  • 12
    This seems pretty convincing! I'll go ahead and accept it. Sep 27, 2022 at 15:36

Seem pretty obvious that the Wizard, being a fake himself, could only give fake gifts -- he sort of says this. The scarecrow obviously did have brains and perhaps armed with a belief in his own abilities would eventually become genuinely educated. Of course the Tin Man had a heart or he would not have wanted one. The Cowardly Lion just, like the others, needed to believe in himself.

Dorothy was given a real magical gift because the witches of Oz were magical. Why they put up with the Wizard I'll never know; perhaps the Good Witch was just being kind or needed someone to rule Oz although she would step in if necessary as she indeed did in Dorothy's case.

  • 19
    this answer is more to the point of the story. The wizard is a fraud and what people needed was confidence. This particularly fits the written work behind the film, which I believe was a commentary against the gold standard for US currency (the gold road to the house of green is a hollow fake, use confidence = fiat trust instead)
    – Mike M
    Sep 27, 2022 at 14:46
  • 8
    @MikeM The monetary policy thesis is a really interesting theory, and one I hadn't read before, but it doesn't seem to be a mainstream or confirmed interpretation. Sep 27, 2022 at 17:54
  • 1
    @dan04: yup: it had to stored, transported and guarded and then stored again. Probably had to be counted and authenticated (at least weighed in those less sophisticated days). I bet the total cost was easily 1 percent of value.
    – releseabe
    Sep 28, 2022 at 0:29
  • 1
    thanks, @JeffBowman -- that, in the link, does sound like what I learned a long time ago and I see it may have been more clever than well-supported
    – Mike M
    Sep 28, 2022 at 18:07
  • 1
    To back up that that it was well known the wizard was a fraud, the lyrics to America's "Tin Man" from the 1970's has the line, "But Oz never did give nothing to the Tin Man; That he didn't, didn't already have."
    – JS.
    Sep 29, 2022 at 1:44

I don't think it's a invalidation of the scarecrow. I think it's saying in fantasy land that math is fantasy too. Or it's being not serious about math, when the moral of the story was the part that was supposed to shine.

  • 4
    perhaps, but the other answers make a good point: it's meant to invalidate the Wizard's powers.
    – Luciano
    Sep 28, 2022 at 8:27

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