circle

Examples of circle in the following topics:

• Circles as Conic Sections

  • You've known all your life what a circle looks like.
  • But what is the exact mathematical definition of a circle?
  • Hence, the definition for a circle as given above.
  • Radius: a line segment joining the center of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.
  • Tangent: a straight line that touches the circle at a single point.

• Introduction to Circles

  • The equation for a circle is an extension of the distance formula.
  • The definition of a circle is as simple as the shape.
  • Since we know a circle is the set of points a fixed distance from a center point, let's look at how we can construct a circle in a Cartesian coordinate plane with variables $x$ and $y$.
  • Now that we have an algebraic foundation for the circle, let's connect it to what we already know about some different parts of the circle.
  • The circumference is the length of the path around the circle.

• The Circle-of-Fifths Progression (In Minor)

• Defining Trigonometric Functions on the Unit Circle

  • In this section, we will redefine them in terms of the unit circle.
  • Recall that a unit circle is a circle centered at the origin with radius 1.
  • The coordinates of certain points on the unit circle and the the measure of each angle in radians and degrees are shown in the unit circle coordinates diagram.
  • We can find the coordinates of any point on the unit circle.
  • The unit circle demonstrates the periodicity of trigonometric functions.

• Radians

  • An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation.
  • The length of the arc around an entire circle is called the circumference of that circle.
  • One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle.
  • A unit circle is a circle with a radius of 1, and it is used to show certain common angles.
  • An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian.

• Introduction to Ellipses

  • Then we can write the equation of the circle in this way:
  • In this equation, $r$ is the radius of the circle.
  • A circle has only one radius—the distance from the center to any point is the same.
  • To change our circle into an ellipse, we will have to stretch or squeeze the circle so that the distances are no longer the same.
  • First, let's start with a specific circle that's easy to work with, the circle centered at the origin with radius $1$.

• Applications of Circles and Ellipses

  • Therefore the equation of this circle is:
  • The center of the circle can be found by comparing the equation in this exercise to the equation of a circle:
  • The radius of the circle is $r$.
  •  The leftmost point on the circle is $(-3,-8)$.
  • The radius of the circle is $r$.

• Key Signatures

  • The circle of fifths is an illustration that has been used in music theory pedagogy for hundreds of years.
  • As you might have noticed, the circle of fifths is so named because each note in the circle is a perfect fifth away from its neighboring notes.
  • The most common usage for the circle of fifths is to help determine key signatures.
  • In Western harmony, the circle of fifths is useful for identifying common chord progressions.
  • Compare the minor key circle of fifths below with the major key circle of fifths above, and you'll see the remaining relative key pairs.

• Angular Position, Theta

  • The angle of rotation is a measurement of the amount (the angle) that a figure is rotated about a fixed point— often the center of a circle.
  • We know that for one complete revolution, the arc length is the circumference of a circle of radius r.
  • The circumference of a circle is 2πr.
  • Because there are 360º in a circle or one revolution, the relationship between radians and degrees is thus 2π rad=360º, so that:
  • The radius of a circle is rotated through an angle Δ.

• Related Keys

  • The circle of fifths is a way to arrange keys to show how closely they are related to each other.
  • This puts them in the same "slice" of the circle.
  • The keys that are most distant from C major, with six sharps or six flats, are on the opposite side of the circle.
  • The circle of fifths gets its name from the fact that as you go from one section of the circle to the next, you are going up or down by an interval of a perfect fifth.
  • Since going down by a perfect fifth is the same as going up by a perfect fourth, the counterclockwise direction is sometimes referred to as a "circle of fourths".