Drum Making

AN INTRODUCTION TO CIRCLES

The number of interesting shapes that there are in Nature is too great to treat all of them, but if there are a few that every person should know a thing or two about , they are: circles, triangles and rectangles, and in 3-d, spheres, cylinders and cubes.

In this lesson, we are going to stick with circles, which could very well be the most important of all of them. We see circles everywhere in Nature. We see them as wheels, in the middle of eyes, and as the cross section of trees. They are important components of early artistic traditions, including Tibetan mandalas and Native American symbols and forms.

But to be able to use circles in doing calculations via equations, we must get a much more rigorous definition of them.

Q1) What is your definition of a circle?

Everyone knows a circle when they see one, but to understand the algebraic definition, it is helpful to consider this more deeply.

A circle may best be defined as all of the points that are at the same distance from a given point, called the CENTER.

So if I were to take a point C as my center, as in Figure 1, and wanted to know all of the points that were 2 units away from the center, those points would make a circle. The distance that I set as the distance away from the center I am interested in is known as the RADIUS.

So now let's say that we wanted to draw a circle on a piece of graph paper like in FIgure 2. Let's further say that we are going to draw a circle whose CENTER is at the ORIGIN, or the point (0,0). We can determine the equation that defines all of the points that are a given distance away by refreshing our understanding of the distance formula.

The distance formula, you may recall, is given as d = ((x1-x2)^2 + (y1-y2)^2). We will consider where this formula comes from HERE, but for now, let's ask the question "What are all of the x-values and y-values that are a distance 2 away from the point (0,0)?" In this case, we could say that one of our points in the distance formula is (0,0), so x2 and y2 are both zero. Now, if I gave you another point (x1,y1), you could tell me the distance between it and (0,0) by plugging in that value and solving for d.

But in this case, I am asking the question: Given a distance, say 2, what are all of the x's and y's that are that distance away from (x2,y2) = (0,0)? So in this case, I would plug in 2 for the distance, and the points that satisfy this are the points (x,y) for which

2 = (x^2 + y^2)^(1/2)