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).