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    $\begingroup$ The reason you add angles is because that's a group homomorphism from $\mathbb{R}$ to the rotation group; you don't have to discount the possibility of multiplying angles to say that. $\endgroup$ – Qiaochu Yuan Jun 26 '11 at 23:22
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    $\begingroup$ Multiplying angles would give some special meaning to an angle of 1 radian. I can't think of any sort of geometric construction in which an angle of 1 radian arises naturally. $\endgroup$ – Corey Jun 26 '11 at 23:59
  • $\begingroup$ @QiaochuYuan, by the way, angles can also be subtracted: this is another way to form a field in $\mathbb{C}$, by complex conjugate multiplication. Is this operation also a group hom? $\endgroup$ – alancalvitti Sep 19 '12 at 18:00
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    $\begingroup$ Solid angles, measured in steradians. $\endgroup$ – Abhimanyu Pallavi Sudhir Jun 8 '13 at 5:50
  • $\begingroup$ Could someone explain what @QiaochuYuan said in simple terms please? $\endgroup$ – user1717828 Apr 27 '16 at 10:27



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    $\begingroup$ Why would the product of two angles be measured in radians? See Gerry's answer; square-radians are not meaningless. $\endgroup$ – user6701 Jun 27 '11 at 7:13
  • $\begingroup$ The question was whether angles are ever multiplied. If I understand your answer correctly, it argues that multiplication of angles wouldn't make sense if angles were closed under multiplication. Apparently angles are multiplied in some areas, and their product is not an angle. $\endgroup$ – user6701 Jun 27 '11 at 12:53
  • $\begingroup$ Bingo. This is the I-should-have-thought-of-that answer that I was looking for. I have up-voted this answer, and accepted it. I love MSE! $\endgroup$ – Mike Jones Jun 27 '11 at 21:44


  • $\begingroup$ I was wondering why no one mentioned Solid Angle and Steradians except for you. Is it not a topic in High School Physics everywhere? $\endgroup$ – Ravi May 9 '12 at 20:57
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    $\begingroup$ Steradians do measure solid angles, but I have only ever seen trig functions of angles actually multiplied together, even in L'Huilier's formula it is tangents of angles that are multiplied together. This preserves the modular congruence that leftaroundabout mentions. $\endgroup$ – robjohn Sep 20 '12 at 11:47



While it's normal in mathematics to eliminate the units of Radians, it will be useful to remember for this example that the radian is a meter per meter (m/m).

Radians are used to represent the planar (plane) angle. But there are other kinds of angles worth considering. One, worth considering enough, has it's own associated unit. That angle is the solid angle, and the unit is the steradian, with dimensions (m²/m²). While of course one could divide an area by area to arrive at this value, it's also plausible, and sometimes necessary, to have to multiply two angles.

Multiplying angles, like any other dimension, results in a new dimension. As such, it's very different from multiplying universal integers, or other universal numerics.



And others, but I can't add the links yet. :/

  • $\begingroup$ excellent point (about m/m) $\endgroup$ – Mike Jones Jun 22 '15 at 12:41
  • $\begingroup$ @MikeJones, if it helps, I've usually seen solid angles in regards to light. Lumens and Lux are both defined in terms of Steradians (lux by proxy). It makes a lot of sense when you start to consider that the "beam" emitted from a flashlight or other lightsource is more complicated than a planar angle. That might be a really good example for your students. Much like a line becomes a square, an "angle" becomes a "beam". $\endgroup$ – Patrick Kelly Jun 22 '15 at 14:32

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