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Geometry: Circles

Introduction to Circles



A circle is the set of all points equidistant from a given point. The point from which all the points on a circle are equidistant is called the center of the circle, and the distance from that point to the circle is called the radius of the circle. A circle is named with a single letter, its center. See the diagram below.

Figure %: A circle

The circle above has its center at point C and a radius of length r. By definition, all radii of a circle are congruent, since all the points on a circle are the same distance from the center, and the radii of a circle have one endpoint on the circle and one at the center.

All circles have a diameter, too. The diameter of a circle is the segment that contains the center and whose endpoints are both on the circle. The length of the diameter is twice that of the radius. Therefore, all diameters of a circle are congruent, too.

Figure %: The diameter of a circle

Keep in mind that an infinite number of radii and diameters can be drawn in a circle. Although they are all congruent, they are not the same. Sometimes a strategically placed radius will help make a problem much clearer. Likewise, diameters can be drawn into a circle to strategically divide the area within the circle. Each of these techniques is prevalent in geometric proofs, and each is based on the facts that all radii are congruent, and all diameters are congruent. However, their position when drawn makes each one different.