REVIEW OF CIRCLES (and related concepts)

• Let $\,C := (h,k)\,$ be a point; let $\,r\,$ be a nonnegative real number.
  Using set-builder notation: $$ \begin{align} &\cssId{s13}{\text{the circle determined by $\,C\,$ and $\,r\,$}}\cr\cr &\qquad \cssId{s14}{=\{\ (x,y)\ |\ \text{the distance from $(x,y)$ to $(h,k)$ is $\,r$ } \ \}} \cr\cr &\qquad \cssId{s15}{= \{\ (x,y)\ |\ \sqrt{(x-h)^2 + (y-k)^2} = r\ \}}\cr\cr &\qquad \cssId{s16}{= \{\ (x,y)\ |\ (x-h)^2 + (y-k)^2 = r^2\ \}} \end{align} $$
• If $\,r = 0\,,$ the circle degenerates to a single point—its center.
  In this case, it is called a ‘point circle’.
• The interior of the circle with center $\,(h,k)\,$ and radius $\,r\,$ is: $$ \cssId{s20}{\{(x,y)\ |\ (x-h)^2 + (y-k)^2 < r^2 \,\}} $$ A circle does not include its interior; it is the boundary (of the interior) only.
• In particular, a (non-point) circle does not include its center.
• See Equations of Circles for more information.

1. the distance from the center to any point on the circle
2. a line segment from the center to a point on the circle

• ‘radii’ (pronounced ray-dee-eye)
• ‘radiuses’

1. twice the radius (a nonnegative number)
2. a line segment passing through the center of a circle,
   with both endpoints on the circle

• In algebra, precalculus, calculus, and life (!) contexts, the words radius and diameter typically refer to numbers (distances).
• In geometry contexts, the words radius and diameter often refer to line segments.
• From context, it should always be clear whether the words radius and diameter refer to numbers (possibly with units) or line segments.

• SYMBOL USED FOR PI:
  The symbol ‘$\,\pi\,$’ is a lowercase letter in the Greek alphabet.
  It is spelled ‘pi’ (no ‘e’ on the end).
  It is pronounced ‘pie’ (as in apple pie or cherry pie).
• HANDS-ON UNDERSTANDING OF $\,\pi\,$:
  • Grab any circular object (like a plate).
  • Measure its circumference and diameter.
    (Use a string or a flexible ruler.)
  • Be sure to use the same units for both measurements.
  • Take the circumference and divide by the diameter—you'll get a number a bit more than $\,3\,$!
  • Repeat with a different circular object, then another (until bored).
  • Any differences in your calculations—as you use different size circles—are due only to measurement errors.
• DEFINITION OF PI:
  For circles, the ratio of circumference to diameter is always the same! This was known by Babylonians and Egyptians at least 4000 years ago.
  
  By definition, $\,\pi\,$ is the ratio of circumference to diameter in a circle.
  Leonhard Euler began the use of the symbol ‘$\,\pi\,$’ for this ratio in 1737. $$ \cssId{s72}{\pi = \frac{\text{circumference}}{\text{diameter}}} $$ The formula for the circumference of a circle follows immediately: $$ \cssId{s74}{\text{circumference} = \pi(\text{diameter})} $$
• PI IS IRRATIONAL:
  The number $\,\pi\,$ is irrational (not rational). This means:
  • it has no representation as a ratio of integers;
    a common rational approximation is $\displaystyle\,\pi\approx\frac{22}{7}\,$
  • it has an infinite, non-repeating decimal expansion;
    a common decimal approximation is $\,\pi\approx 3.14\,$
  • for more decimal places of pi, look to the right,
    or go to Wolfram|Alpha and type in ‘pi’
• PRACTICAL INTERPRETATION OF PI:
  Keep in mind what all this is saying! In any circle, if you walk all the way around, you'll travel just over three times further than if you walk straight across (passing through the center) to the opposite side.

CENTRAL ANGLE OF A CIRCLE

• By convention, ‘central angle’ refers to the smaller of the two angles determined by two radii.
• That is—unless otherwise requested—a central angle is between
  $\,0^\circ\,$ and $\,180^\circ\,.$
• If you want the bigger central angle, ask for the ‘reflex central angle’.
• The name ‘central angle’ is appropriate, since the vertex of a central angle is the center of a circle.