# Circle

Pronunciation: /ˈsɜr.kəl/ Explain Figure 1 - Using a compass to draw a circle.

A circle is all points in a plane that are equidistant from the center point. When using a compass to draw a circle, the point of the compass is the center of the circle, and the stylus marks all points that are the same distance from the center.

### Properties of a Circle

Click on the blue points in manipulative 1 and drag them to change the figure. Double click on the manipulative to open it in a full-screen window.

 Click on the blue and orange points and drag them to change the figure. Can a chord also be a diameter? Can a tangent line also be a radius? Manipulative 1 - Parts of a Circle Created with GeoGebra.
• The center of a circle is the point from which all points of the circle are equidistant.
• A radius of a circle is a line segment from the center of the circle to one of the points on the circle.
• A diameter of a circle is line segment from one point on the circle to the opposite side through the center of the circle. The length of a diameter is twice the length of a radius (d = 2r).
• The circumference of a circle is the edge of the circle. Circumference can also refer to the length of the edge of the circle.
• A chord of a circle is a line segment from any point on the circle to any other point on the circle. See Chord.
• An arc is a portion of the circumference of the circle.
• A sector is a portion of a circle between two rays radiating out from the center of the circle.
• A segment is a portion of a circle cut off by a chord of the circle.
• A tangent of a circle is a line that touches the circle at exactly one point. The point where the tangent line touches the circle is called the point of tangency.
• A tangent segment of a circle is a segment of a line tangent to a circle extending from the point of tangency to a point of intersection with a ray extended from the center of the circle.

#### Understanding Check

1. Is a diameter also a chord? Try moving the endpoints of the chord in figure 2 on top of the endpoints of the diameter. Yes. No.
2. Does each circle have exactly one diameter? Try moving the endpoints of the diameter in figure 2. Yes. No.
3. What is the relationship between the measure of a radius and the measure of a diameter of a circle? The measure of a radius has no relationship to the measure of a diameter. The measure of a radius is 1/4 the measure of a diameter. The measure of a radius is 1/2 the measure of a diameter. The measure of a radius is exactly the same as the measure of a diameter.

### Formulas Related to a Circle

Definitions of Variables
VariableRepresents
ddiameter
ccircumference
aarea
hx-coordinate of the center of a circle
ky-coordinate of the center of a circle
DescriptionEquation
Diameter of a circled = 2r
Circumference of a circlec = 2πr
Area of a circlea = πr2
Equation for a circle in Cartesian coordinates(x - h)2 + (y - k)2 = r2
Equation of a circle with center at the origin in Polar coordinatesr = r0
Table 1: Formulas relating to a circle

### Center-Radius Equation of a Circle

 Click on the black points on the slider and drag them to change the figure. Note how the equation changes as the sliders are moved. Can you place a circle with radius 1 at the origin, (0,0)? Manipulative 2 - Equation of a Circle Created with GeoGebra.

The center-radius equation of a circle is (x - h)2 + (y - k)2 = r2. In this equation the point (h, k) is the coordinates of the center of the circle, and r is the radius of the circle.

### How to Plot a Circle

For this demonstration use the center-radius equation for a circle centered at (-1, 2): (x + 1)2 + (y - 2)2 = (1.5)2.

StepDiagramDescription
1 Since x - h = x + 1, h = -1. Since y - k = y - 2, k = 2. So the center of the circle is at the point (-1, 2). Plot the point (-1, 2).
2 Now draw the two points that are on the circle to the right and left of the center. Since the radius is 1.5, the left point will be at (-1 - 1.5, 2) = (-2.5, 2) (the x-coordinate of the center of the circle minus the radius). The right point will be at (-1 + 1.5, 2) = (0.5, 2) (the y-coordinate of the center of the circle plus the radius).
3 Now draw the two points that are on the circle that are on the top and bottom of the center. Since the radius is 1.5, the top point will be at (-1, 2 + 1.5) = (-1, 3.5) (the y-coordinate of the center of the circle plus the radius). The bottom point will be at (-1, 2 - 1.5) = (-1, 0.5) (the y-coordinate of the center of the circle minus the radius).
4 Now, draw a circle through the four points.
1. McAdams, David E.. All Math Words Dictionary, circle. 2nd Classroom edition 20150108-4799968. pg 33. Life is a Story Problem LLC. January 8, 2015. Buy the book

• McAdams, David E.. Conic Section. allmathwords.org. Life is a Story Problem LLC. 6/27/2018. http://www.allmathwords.org/en/c/conicsection.html.

### Educator Resources

• Circumference and More Geometry. NASA LaRC Office of Education. 2/18/2010. http://www.archive.org/details/NasaConnect-Eom-CircumferenceAndMoreGeometry/index.htm.

McAdams, David E. Circle. 4/13/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/circle.html.

### Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)