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Angles as rotations compared to angles between rays

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Drag the slider or click the "play" triangle in the lower left corner to animate this diagram. To watch the animation more slowly, click the point on the slider and use the right arrow key.

In Euclidean geometry, an angle is generally considered as the amount of opening between two rays. Usually it is considered as the smallest angle (less than 180°.)

In transformational geometry, an angle is considered as a rotation, often measured in turns and fractions of a turn. In this diagram, ray a is fixed, and ray b rotates. An angle of more than 1 turn means ray b has gone around more than once. The convention for higher math (trigonometry, calculus) is that positive rotations are counterclockwise. Then negative angles are clockwise rotations.

This diagram shows positive (counterclockwise) rotations from 0 turns to 5 turns, and the resulting angle between the rays.

Further details: the slider angle changes in increments of 1/10 turn (36 degrees.) To get another specific angle,
Double click on the geometry window to open the full program
Type your value for the parameter "turns" in the input window, as a fraction of a turn. For example, to get at 37 degree angle, type


Exercises to check your understanding:

List all the rotations that result in a 36° angle. Use + for counterclockwise rotations, - for clockwise, and ... to show a continuing pattern.










Answer, in turns: ..., -2.9, -1.9, -0.9, 0.1, 1.1, 2.1, 3.1, ... and ..., -2.1, -1.1, -0.1, 0.9, 1.9, 2.9, 3.9, ...

Susan Addington, Created with GeoGebra

Updated May 15, 2010


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