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In a math test a question was to draw a $405^\circ$ angle. Is it formally correct to say draw an angle as I think that in geometry, an angle has just some formal definition. So what is the connection between the formal definition and the drawing? And how do one draws angles over $360^\circ$?

Student
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    Basically like this http://www.icoachmath.com/image_md/Quadrantal-Angle2.jpg – graydad Mar 18 '15 at 15:18
  • Ha! Now we know that in your native language, $405$ starts with a vowel. So it's not English, French, German, Hungarian, Russian, Mandarin Chinese, Thai, Japanese,... Does anybody have any suggestions? – TonyK Mar 18 '15 at 15:19
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    The question was in Finnish matriculation examination. In Finnish, 405 is neljäsataaviisi. I might have problems with English grammar. – Student Mar 18 '15 at 15:24
  • @Student In English, we use an before vowel sounds, and a otherwise. Thus: an angle, but a $405˚$ angle ("a four-hundred-and-five-degree angle"). In any case, your English is very clear. – Théophile Mar 18 '15 at 15:32
  • That's my theory ruined then! I was thinking Indonesian/Malaysian "an empat ratus lima $^\circ$ angle". And yes, your English seems excellent to me; I was reading too much into your little slip-up. – TonyK Mar 18 '15 at 15:33

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It depends on how you think about angles. You can either agree that $405^\circ$ is exactly the same as $45^\circ$. -- This is how mathematicians usually think about it. Or you can think about it as $1$ complete rotation ($360^\circ$) and then an additional $45^\circ$. -- This is how engineers usually think about it.

The way you draw it depends on which of the two ways above you think about it, but the angle should start on the positive $x$-axis and end on the ray positioned at a $45^\circ$ angle counterclockwise from that position either way you think about it.

  • Mathematicians think of it both ways, connected by a thing we call "winding number". – David Wheeler Mar 18 '15 at 15:21
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    Obviously I was being overly simplistic (because Student is probably in secondary school). –  Mar 18 '15 at 15:21
  • I'd say that both mathematicians and engineers think both ways. To be precise, it's the difference between the abelian group $\mathbb{R}\setminus\mathbb{Z}$ acting on itself and $\mathbb{R}$ acting on it. – lisyarus Mar 18 '15 at 15:24
  • @Bye_World It's all good, angles are intimately related to rotation, and it's useful to know if "full turns count". – David Wheeler Mar 18 '15 at 15:25