# How to draw an $405^\circ$ angle?

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 $360^\circ$?

• Basically like this icoachmath.com/image_md/Quadrantal-Angle2.jpg Commented Mar 18, 2015 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? Commented Mar 18, 2015 at 15:19
• The question was in Finnish matriculation examination. In Finnish, 405 is neljäsataaviisi. I might have problems with English grammar. Commented Mar 18, 2015 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. Commented Mar 18, 2015 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. Commented Mar 18, 2015 at 15:33

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.
• 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. Commented Mar 18, 2015 at 15:24