Radian and Steradian

In this post, we'll discuss the concept of angles and the methods for measuring them. This will be a journey based not on abstractness but rather on intuitive rigor. By this, I mean, we shall develop the concepts from the known fundamentals, and at each step, ask if we can convince ourselves that what we are doing makes sense to the common intellect. We'll start by defining an angle and studying its geometry. Then we'll try to understand the different units employed to measure this with an emphasis on radian. Finally, we'll also introduce the concept of the solid angle and steradian. So let's begin our journey!

An Angle:

Defining an angle is as difficult as defining a line. Ask yourself, "Can I write a statement that completely encompasses the concept of line?" You may call it, like Euclid said, a 'breadthless length'. But then what do you mean by length? Can you really explain something like length without the concept of a straight line? How do you define what is straight? And these are not dumb questions. It is simply a limitation of our vocabulary. The problem is not in understanding what a line is, but rather teaching it to another man with just words.

Terms like line and angle cannot be taught by words; they must be made to be felt by the learner. The learner needs to see it, needs to sense it, and only then shall he truly understand what the line is. The line is then not just an expression but becomes an experience. It is this experience that we also seek to understand the angle. This experience may begin with intuition, but it must also incorporate rigor sooner or later. Rigor helps us filter out the medley of thoughts and have clarity in our ideas. We shall proceed with this motive to understand the angle.