It’s Professor Dave, let’s talk about circles.
We’ve been talking about polygons for a while, which are shapes made of line segments.
There are plenty of other shapes to talk about where the edges are not straight lines, but
are curved in some way.
The simplest of these shapes, or really, the simplest of all shapes, is the circle.
A circle is simply the set of all points that are a particular fixed distance from some
central point.
So unlike triangles and other polygons where we have to define all kinds of side lengths
and angles, to get a circle we only need to define two things.
These are the center of the circle, and then a specific distance from that point, like
five centimeters.
If we draw all the points in this plane that are five centimeters away from this central
point, we get a circle.
The distance from the center to any point on the circle is called the radius of the
circle, and any line segment connecting the center and some point on the circle can be
referred to as a radius.
Other line segments of note include chords, which are line segments with both endpoints
on the circle.
A chord that passes through the center of the circle is called a diameter, and we will
note that the length of any diameter of a circle is always exactly twice the length
of its radius.
Lastly, any part of a circle defined by two endpoints is called an arc.
If this arc becomes the whole circle, which represents the distance around the circle,
or the perimeter of the circle, this is called the circumference.
Just the way that the diameter and radius are always in a two to one ratio, the circumference
and the diameter also have a special ratio, and that ratio is represented by the number
pi.
This is an irrational number, which as we discussed before, is a number that can’t
be expressed as the ratio of two fractions.
This means that it has a decimal representation with digits that go on forever, without any
kind of repeating pattern.
While this is sufficiently mind-boggling when you think about it, for our meager purposes
we can just estimate pi as being three point one four, as that will give us values that
are precise enough to be meaningful.
So the equations we need to know are as follows.
D equals two R, and C equals pi D, or in terms of R, C equals two pi R, since one D is two
R’s.
Knowing this, if we are given any of these values for a circle, we can find the others.
For example, if the radius is five, what is the circumference?
Well, we can double the radius to get a diameter of ten, and then multiply by pi to get a circumference
of ten pi.
Going backwards, if we have a circumference of eighteen pi, we can find a diameter of
eighteen, and a radius of nine.
If we draw several radii on one circle, we will demarcate several sectors of that circle.
A sector is a part of a circle enclosed by two radii and the arc that connects them,
like a slice of pizza.
Within this sector there is a central angle, and the angles of all the sectors in a circle
will always add up to three hundred sixty, or one full revolution.
There will be plenty more to discuss regarding circles later, but for now, let’s check comprehension.