**History**

The study of the circle goes back beyond the
recorded history. The invention of the wheel is a fundamental
discovery of properties of a circle. The Greeks considered the
Egyptians as the inventors of geometry. Ahmes, who is a scribe
and the author of the Rhind papyrus, gives a rule for determining
the area of a circle that corresponds to 256/81 or approximately
3.16. Thales found the first theorems relating to circles around
650 BC. The Euclid's Book III, *Euclid's Elements* set to
work properties of circles and problems of inscribing polygons.
One of the problems of Greek mathematics was the problem of finding
a square with the same area as given circle. Several of the 'famous
curves' were first attempted to solve this problem. Anaxagoras
in 450 BC is the first recorded mathematician to study this problem.

**Lesson I: Definition and geometric construction
of a circle**

**Definition**

A circle, as a curve, is the set of points
equidistant from a given point. That is, a circle is a curve consisting
of all those points of a plane that lie at a fixed distance a
from a particular point, called the *center* of a circle,
in the plane. The distance a from the center is called the *radius*
of the circle and the perimeter of a circle is called the *circumference*.
In this sense, all points on the circumference of a circle are
equidistant from its center. *Diameter *is the longest segment
through the center and is twice the radius, , and
*circumference *is given by .

If the center is at the origin of the Cartesian coordinate system:

If the center is not at the origin:

**How is a circle obtained from a conic section?**

A circle is obtained by the intersection of a cone with a plane perpendicular to the cone's symmetry axis. See the following figure:

**Student Work**:
Have students construct their own circle on GSP and change the
center and the radius.

**Lesson II: Introduction to the algebraic
representation of a circle**

**Equation from definition**

Let's, suppose, in rectangular coordinate plane, take a point C (p, q) as a fixed point and the distance from the point (p, q) is a. Since r is the distance, we don't need to specify that a is a positive number. We are looking for the equation of the circle, which is satisfied with such conditions given. If you want to find a point P (x, y) on the circle, it can be traced in the Cartesian plane. Now you have:

1) If the center lies on the origin of the plane:

2) If the center is not concurrent with the origin:

The above formula is considered as the standard equation of a circle. The fixed point C (p, q) is the center and r is the radius of this circle, according to the definition. Given any equation of this form, you can guess the geometric figure corresponding to this equation is a circle with the center (p, q) and radius r. If you have the origin (0, 0) as the center, that is you put 0 into a and b, you get the equation

**Prerequisite**:
In order to draw the final equation form mathematically from the
word sentence, you should first know the definition of distance
as a mathematical term and express it with mathematical sentence
like the first equation. You also have to understand the Pythagorean
theorem to get the distance between any two points.

**Lesson III: Analytic Equation of a Circle**

Let's suppose the center of a circle is not at the origin (0, 0) (See Figure 5.). We already have the equation and can expand:

The last formula is called a general quadratic equation of a circle. In fact, this formula is a general equation for all conic section curves. If you change the value of A, B, and C, you may get a circle, a parabola, an ellipse, or a hyperbola. On the contrary, given the general quadratic equation, if C is positive, the equation represent a circle having the center (p, q) and radius expressed in terms of a square root.

**Lesson IV: Parametric and Polar Equations
of a Circle**

We can get a circle in polar coordinates. When the center is at the origin (0, 0), in figure 4, according to definition of a circle, you have:

The *parametric equations* for a circle
of radius r are:

The *polar* equation for a circle having
the center at (0, 0) and the radius a is:

For a circle having the center at (p, 0) and the radius a, the polar equation is:

Similarly, the polar equation for a circle with the center at (0, q) and the radius a is:

**Lesson V: Properties of a circle**

The ratio of circumference to diameter is always
constant, denoted by p , for a circle with the radius a as the size of the
circle is changed. Let's define d as *diameter* and c as
*circumference*. Then, as observed, since

the ratio is:

**Exploration: Apollonius Circles**

**Definition**: The
locus of a point P whose distances from two fixed points A and
A' are in a constant ratio 1 : m , so that

**Investigate** the Apollonius
Circles.

**References**

Coxter, H. S. M. (1969). *Introduction to
Geometry.* New York: John Wiley & Son, Inc.

http://mathworld.wolfram.com/Circle.html