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Top 10 things to know about circles

  1. A circle is the set of points in a plane that are equidistant from a given point $O$ called the center.
  2. The distance $r$ from the center is called the radius, and the point $O$ is called the center. Twice the radius is known as the diameter, $d = 2r$.
  3. The angle a circle subtends from its center is a full angle, equal to $360^\circ$ or $2\pi$ radians.
  4. The perimeter $C$ of a circle is called the circumference, and is given by $C = 2\pi r = \pi d$.
  5. The area of a circle is given by $A = \pi r^2$.
  6. A central angle is an angle with endpoints $A$ and $C$ located on a circle's circumference and vertex $O$ located at the circle's center.
  7. For an inscribed angle and central angle with the same endpoints,

    \begin{displaymath}\angle AOC = 2 \angle ABC \end{displaymath}

  8. A tangent to a circle at a point $p$ is perpendicular to the radius $OP$.
  9. The circumcircle is a triangle's circumscribed circle, i.e., the unique circle that passes through each of the triangle's three vertices. The center $O$ of the circumcircle is called the circumcenter, and the circle's radius $R$ is called the circumradius. A triangle's three perpendicular bisectors (i.e. the line perpendicular to the side from the midpoint of the side) meet at $O$.
  10. The incircle is the inscribed circle of a triangle , i.e., the unique circle that is tangent to each of the triangle's three sides. The center $I$ of the incircle is called the incenter, and the radius $r$ of the circle is called the inradius. The incenter is the point of concurrence of the triangle's angle bisectors.

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Steven R. Dunbar
Department of Mathematics and Statistics
University of Nebraska-Lincoln
Lincoln, NE, 68588-0323 USA
email: sdunbar@math.unl.edu
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