## Summary

IVocabulary and parts of a circleIILines, angles and segments in a circleIIIAreas & circumference in a circleIVCircles in the coordinate plan and equation of a circle## Vocabulary and parts of a circle

### Circle

A circle is the collection of points of equal distance from a central point.

The following graphic contains a circle. The center of the circle is labeled as C. The points on the circle are all of distance r from the center of the circle.

### Radius

The radius of a circle is the distance from a point on the circle to the center of the circle.

### Diameter

The diameter of a circle is the distance from one point on the circle through the center and to another point on the circle.

If d is the diameter of a circle and r is the radius of the circle then:

** d=2r **

### Arc

An arc is a connected portion of a circle.

### Chord

A chord of a circle is a line segment which connects two points on the circle. A chord **does not** have to contain the center of the circle.

A chord that passes through the center of a circle is a diameter.

### Tangent of a circle

A tangent of a circle is a line which touches a circle in exactly one point.

## Lines, angles and segments in a circle

### Interior angle

An angle formed by two chords of a circle which share a common point is called an interior angle of the circle.

### Central angle

An angle formed by two radiuses of a circle is called a central angle.

### Constant interior angle

If \alpha is the measure of an interior angle of a circle and \beta is the measure of the central angle which has the same endpoints as the interior angle, then:

** \beta=2\alpha **

### Constant interior angle theorem

Any two interior angles of a circle with the same endpoints are congruent.

### Interior angle which intercepts a diameter.

An interior angle whose endpoints form a diameter of a circle measures 90^\circ.

### Exterior angle

An exterior angle of a circle is an angle formed by two line segments which meet at a point outside of the circle, but whose endpoints are on the circle.

### Exterior angle theorem

Consider the following figure:

Then:

** \alpha=\dfrac{\gamma-\beta}{2} **

Consider the following figure:

Solve for \alpha :

\alpha=\dfrac{160^\circ-30^\circ}{2}=\dfrac{130^\circ}{2}=65^\circ

## Areas & circumference in a circle

### Circumference

The circumference of a circle is the length around the circle.

### Ratio of circumference and diameter

There is a real number, denoted by \pi, such that if a circle has radius r, a diameter d=2r, and a circumference of C then

** C=2\pi r=\pi d **

The number \pi is irrational and is approximately:

\pi\simeq 3.14\ 159

A circle with a diameter of 3 has a circumference of 3\pi.

### Area of a circle

The area of a circle of radius r is:

** \pi r^2 **

A circle of radius 3 has an area of:

\pi\left(3\right)^2=9\pi

## Circles in the coordinate plan and equation of a circle

### Equation of a circle

In an orthonormal coordinate system, the equation of the circle of radius r and center \left(a,b\right) is:

** \left(x-a\right)^2+\left(y-b\right)^2=r^2 **

Consider the following equation:

x^2+y^2=9

The set of points whose coordinates \left(x,y\right) satisfy the above equation is the circle of radius \sqrt{9}=3 centered at \left(0{,}0\right).