The Conics

By: Diana Brown

Day Two:

Circles

Definition of a Circle:

 

A circle is the set of all points in a plane equidistant from a fixed point called the center point. The distance from the center to the circle is called the radius.

 

We can derive the formula from the distance formula:

 

If we square both sides and simplify we get:

x² + y² = r².

 

This is the equation of a circle with its center at (0, 0).

 

Let’s graph an equation of a circle in Graphing Calculator 3.0:

 

We will graph the equation:          x² + y² = 9.

 

 

 

 

Notice the radius is 3.  Since r² = 9; r = 3.

 

Let’s construct a circle with this same equation in GSP.

 

 

 

 

 

 

Notice that the radius constructed is also a length of 3, which by definition no matter where we move the point on the circle it should always be 3 units from the center in the above example.  Click here to open the above GSP file to move the point around the circle.

 

 

A translation of the circle equation becomes:

 

(x – h)² + (y - k)² = r²

 

With center at (h, k) and radius r

 

Here are some examples:

 

 

Equations:

 

 

Sample Practice problems for students

 

1)      Find the center, radius and graph the equation:  (x – 5) ² + (y + 3) ² = 25

2)      Find the center, radius and graph the equation:  x ² +  y ² = 100

3)      Graph a general equation of a circle with center at (0, 0) and various values of r.  Describe what happens.

4)      Do the same as above but leave r as a constant and try various centers.  What do you notice?

 

Definitions Related to Circles:

arc: a curved line that is part of the circumference of a circle
chord: a line segment within a circle that touches 2 points on the circle.
circumference: the distance around the circle.
diameter: the longest distance from one end of a circle to the other.
origin: the center of the circle
pi (pi): A number, 3.141592..., equal to (the circumference) / (the diameter) of any circle.
radius: distance from center of circle to any point on it.
sector: is like a slice of pie (a circle wedge).
tangent of circle: a line perpendicular to the radius that touches ONLY one point on the circle

 

General Circle Formulas:


Diameter = 2 x radius of circle

Circumference of Circle = PI x diameter = 2 PI x radius
 Area of Circle:
    area = PI r2

Length of a Circular Arc: (with central angle theta)
    if the angle thetais in degrees, then length = thetax (PI/180) x r
    if the angle thetais in radians, then length = r x theta

Area of Circle Sector: (with central angle theta)
    if the angle thetais in degrees, then area = (theta/360)x PI r2
    if the angle thetais in radians, then area = ((theta/(2PI))x PI r2

 

 

Go to Circles Investigations (Day Three)

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