Three notions of angle -- By Axelle Faughn

If you stop for a moment to think about the notion of angle, you will realize that what seems like a pretty simple geometric figure of two lines (or rays) intersecting one another, actually has far-reaching ramifications in the world of mathematics. With the help of Mathematical Selfies, we consider three major interpretations of the concept of angles, which we believe must be presented to all children learning about angles in school.

1) Angle as sector

First let us explore the idea of angle as "space in-between two lines", in other words a sector of the plane. This is the original definition of angle which is introduced early on in schools when studying shapes. Children pretty quickly know how to differentiate right triangles from equilateral ones for instance by looking at the positions of the sides with respect to one another. When prompted to illustrate the notion of angles, this is often the one students consider since structures heavily rely upon how angles fit together within a plane.

Picture 1: Angle of best fit

Picture 2: Reaching out into the light

Picture 3: One fifth of 360

Picture 4: Angles in constructions

As in picture 1-4, we can assign a measurement to these angles and categorize them as acute, obtuse or right. Repeated practice with finding all three types of angles in the world around us seems like an important part of internalizing these notions for students in primary grades.

2) Angle as arc length

When more accurate angle measurements come into play, we step into the second notion of angle, which can be thought of as an arc length, or a curved measurement. The use of a protractor as a measuring device illustrates this notion quite nicely since the tool itself has the shape of the arc along which one will read the units.

This aspect of angles can be associated with circular motion and may come with a positive or a negative direction as in trigonometry. There is a very clear connections between arc length and angle measurement established through the definition of radians for measuring angles: if an angle Theta at a given vertex is measured in radians, then the arc length S spanned by this angle along a circle of radius R centered at that vertex is given by S=(Theta)x(R) . In trigonometry that is why in a circle of radius one unit, we can alternatively think of angles as sectors between two rays of the circle, or as distanced arc length of a point traveling on the circle (Picture 5).

Picture 5: The trigonometry of a fan

Picture 6: Copying an angle

Angles as circular motion are also used when copying an angle with the use of the traditional construction tools compass and straightedge (Picture 6). In systems of levers and pulleys, or just hinges on a door (Picture 7) where acute versus obtuse angles determine the closing versus opening of the door, these notions become strategic in gaining mechanical advantage, one of humans' everlasting endeavors.

Picture 7: Using angles for mechanical motion

3) Angle as walking motion 

When giving directions to someone you may find yourself telling them to make a 90 degree left or right turn, or explaining that the road takes a "sharp" right. All these, along with the idea of U-Turns, refer to angles that allow us to locate ourselves in the plane. Following a GPS uses the same types of commands. This notion of angles is less often considered in the mathematics classroom, although certainly not less important in our world of satellites and geo-location devices. Even Pokemon-Go wouldn't work without it! It is therefore worth mentioning here as using the idea of vector angles, in other words the angle established by two directions, wherever these originate from in the plane. Here we dive into the world of equivalence classes. In my geometry class, students have yet to submit illustrations for this particular angle notion, which I take as a hint that I do not emphasize it enough in class.

Picture 8: Octagon dilation

Interestingly enough, angles also play an important role in picture taking. Depending on the angle the photographer positions themselves with respect to their representations, these might change from the object to the selfie based on the angle from which the picture was taken. For instance in Picture 8 illustrating similarity, the viewer knows that a regular octagon is equiangular, however it is not so visually obvious if not facing the shape directly from below.

Finally, any adept photographer knows to use angles for capitalizing on the power of reflections to enhance a picture or create interesting patterns. Picture 9 is only one example of this play with reality using the laws of incidence.

Picture 9: Tree reflection

Picture 10 provides a much more creative illustration of using the mathematics of angles for the purpose of artistic ventures.

            

              Picture 10: Optical incidence, courtesy of Eric Mortemousque