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I just had this terrifying conversation:

What is "triangle" in Japanese?

me: 三角 right?

Then what's 三角形?

me: Uhh... triangle?

I've confirmed both of these translate to "triangle". Apparently they refer to different aspects of the triangle in Japan? Is 三角 a symbol and 三角形 a geometrical form?

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    三角形 is a mathematical (official) term. In a math class, you should call it 三角形.
    – Gradius
    Aug 7, 2012 at 8:05

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It might be something as simple as:

三角　(something that is "triangular" where the focus is having attributes similar to that of triangles ie: three sides, three corners)

三角形　(a polygon that IS a triangle)

For example: 「三角屋根」 is a way to describe a roof that is "triangular" in comparison to other roofs of different shapes. It has attributes similar to those of "triangles".

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    Okay, I see. The 形 means something like it follows all the Euclidean rules. Thank you!
    – Avery
    Aug 7, 2012 at 13:43
  • Honestly speaking, I cannot understand either explanation. Are you claiming that 三角 has focus on three corners whereas 三角形 has focus on three sides? If so, I disagree, but I am not sure if you meant this.
    – Tsuyoshi Ito
    Aug 8, 2012 at 2:42
  • @TsuyoshiIto No, I meant that 三角形 focuses on the polygon as a whole, whereas; 三角 focuses on something that can have similar attributes to a triangle. I'll edit my answer to show that.
    – Chris
    Aug 8, 2012 at 5:09
  • I am not sure what you mean by “something that can have similar attributes to a triangle.” Is it the same as “something that is similar to a triangle”? I do not think that we make a distinction between 三角形 and 三角 depending on whether it is exactly a triangle or just similar to a triangle.
    – Tsuyoshi Ito
    Aug 8, 2012 at 14:18
  • I agree that 三角 can be used to make compound words with the meaning of "triangular" but usually 三角形 cannot. However, as single nouns, 三角 and 三角形 have almost the same meaning. We often use 三角, because it is shorter and easier to say.
    – Gradius
    Aug 8, 2012 at 19:00
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As Gradius said, the mathematical term “triangle” is 三角形, and never 三角. As part of compound words, 三角 also appears; an example is 三角関数 (trigonometric functions).

(As for the use of 三角 in compounds words, I think that there is a general tendency to prefer to two-kanji words than three-kanji words when they are used adjectivally in compound words. See also Chocolate’s comment on the question “What is the difference between 日曜日 and 日曜?” by BabyAzerty.)

As a non-technical word, 三角形 is more formal and 三角 is less formal. I do not recognize any difference in their meanings.

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ya know sometimes what natives say can be misleading and not actually reflective of their intuitions. i think the difference is just as Chris says, "triangle" vs "triangular". i might add though that 三角 can refer to a specific triangle sketched on a piece of paper or blackboard in the sense that you can say "look at this 三角 i just drew, its prettier than yesterday's 三角". but 三角形 doesn't have that power to specify one triangle among others, its referent is just ambiguously the whole class of triangles.

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