Circles

Definitions A circle is the set of all points in a plane that are the same distance from a fixed point called the center of the circle. Note: The center is NOT part of the circle. A radius of a circle is a line segment extending from the center to the circle. A diameter is a line segment that joins two points on the circle and passes through the center. radius center diameter

Naming a Circle A circle is named by its center. The circle at right is called circle O Note: two circles in the same plane with the same center are called concentric circles. O

The word radius (plural: radii) is also used to denote the length of a radius (all radii have the same length). The word diameter is also used to denote the length of a diameter (all diameters have the same length). Note that the diameter of a circle is twice its radius.

Chords A chord is any line segment that joins two points on a circle. Therefore, a diameter is an example of a chord. It is the longest possible chord.

Arcs An arc is an unbroken part of a circle. For example, in the figure, the part of the circle shaded red is an arc.

Naming Arcs A minor arc, like the one in red in the figure, can be named by drawing an arc symbol over its endpoints: AB Sometimes, to avoid confusion, a third point between the endpoints is used to name the arc: APB or BPA The reason for this given two points on a circle, there are two arcs between them (the long way around the circle or the short way). A P B

The arc highlighted in red in the figure would be called Semicircles and major arcs must be named with three (or sometimes more) points. The arc highlighted in red in the figure would be called ABC or CBA. It appears to be a major arc. If we wrote AC then we would be referring to the part of the circle that is not highlighted in red (a minor arc, it seems). A B C

The Measure of an Arc Each arc has a degree measure between 0 degrees and 360 degrees. Therefore it must be greater than 0 and less than 360. A full circle is 360 degrees; a semicircle is 180 degrees. If an arc is a certain fraction of a circle, then its measure is the same fraction of 360 degrees. C A E B D F G H

Arc around the circle The measure around the outside of the circle is called the circumference. It is the distance traveled if you go around the entire arc of the circle.

To draw a circle, we use a compass.

Circumference The Ancient Greeks discovered that there was a special relationship between the circumference of a circle and it’s diameter. Let’s see:

Find the circumference C = 2π r= (2)( )(14) = 88 m C = 2π r= (2)(3.14)(4.5) = 28.26 in C = π d = (3.14)(10) = 31.4 cm

Area of a Circle

Find the area of the circles.

Sector A sector of a circle is a section formed by two radii and a part of the circle called the arc.