How to Discover Pi for Yourself Using Circles

Basic geometry of the circle in a plane

1. 1
   

   You DON'T NEED to memorize formulas for this and that. You will need to use formulas, but that is for another day!
   Begin freshening your understanding of the geometry of the circle in a plane. You know a lot about the point, plane and space, and they're not even defined in the study of geometry, but they are described as they are used.
   • What is a circle? The following information needs to form part of your (basic) understanding of things about circles, but one can learn a lot more as you go along.
   • equidistant - is short for "of equal distance"
   • circle - all the points equidistant, from the center (centerpoint).
   • The following facts relate to but are not part of the circle:
     • center - the point equidistant from any point of the circle,
     • radius - the segment (names the length) between one endpoint at the center and the other end on the circle (it's that "equal distance" mentioned),
     • diameter - the segment (names the length) through the center and between its two endpoints on the circle,
     • segment, area, sector, and included or inscribed shapes within, but not part of, the circle, and
     • circumference - the distance one time around the circle.
       • Yeah, that word is long and odd; so, think of "the distance around circular-fence."

Create a formula first

1. 1
   Discover your circumference formula: The diameter can be curved and placed end to end around the circle, about three times--meaning that: three diameters plus a small fraction of diameter = Circumference. Let's call that C = 3 X d, approximately. Done (that was too easy...), just as you would have had to do originally while discovering circumference about 3000 or 4000 years ago; now you will clean that idea up... In ancient times, math was like a mystical study and your "discovery" was part of the expression of mathematical mysteries.
2. 2
   Absorb that rough, intuitive idea of pi, about 3, and realize it's easily demonstrated that it is not exactly three. Now you will make it more accurate.

Discover pi more exactly

1. 1
   Number four different sizes of circular containers or lids. A globe or ball (sphere) can work also, but it's harder to measure.
2. 2
   Get a non-stretchy, non-kinky string and a meter-stick, yardstick or ruler.
3. 3
   Make a chart (or table) like the following one:
   
   Circumference | diameter | quotient C / d = ?  
   1. __________|________|__________________
   2. __________|________|__________________
   3. __________|________|__________________
   4. __________|________|__________________
4. 4
   Measure accurately around each of the four circular items by wrapping a string snugly around it. Mark the distance one time around it on the string. This is the circumference: it's just like perimeter, but, the perimeter of a circle--the distance around a circle--is called the circumference, not perimeter, usually.
5. 5
   

   How is the perimeter of circles the spark for lighting-up the idea of pi? Here the string has been cut to the length of the circumference.
   Straighten and measure the part of the string that you marked as the distance around the circle. Write down your measurement of the circumference using decimals.
   
   Pin or tape the ends of the string for measuring it accurately (straight and extended to its full measure), since you would have needed to tighten the string around the circular object, so now you would tighten it lengthwise.
6. 6
   Turn the container upside down so you can find and mark the center on the bottom so that you can measure the diameter using decimals (also called decimal-fractions).
7. 7
   

   The diameter "must be" measured very precisely straight across through the center of each circle.
   Measure across each circle exactly through the center of each of the four items with a straight edge measure (meter-stick, yardstick or ruler). This is the diameter.
   • Note: Multiplying two times radius, i.e.: "2 X radius = diameter" is also written as "2r = d".
8. 8
   

   Divide each circumference by the same circle's diameter.
   Divide each circumference by the same circle's diameter. The four division problems of C / d = _____, should be about 3 or 3.1 (or about 3.14 if your measurements are accurate); so what is pi: It's a number. It's a ratio. It relates diameter to circumference. Of course, using precise measurements using dividers, which are similar to a compass can help.
9. 9
   Average the four answers to the division problem by adding those four quotients and dividing by 4, and that should give a more accurate result (for example, if your four divisions gave you: 3.1 + 3.15 + 3.1 + 3.2 = ____ /4 = ____? That's 12.55 / 4 = 3.1375, and can be rounded-off to 3.14).
   
   That's the idea of "pi". The number of diameters that makes the circumference (all the time, so it's constant)... That is the constant "pi". That number of diameters.
   • Also, the radius will fit a little more than 6 (2 times pi) times around a circle, as well as knowing that the diameter goes three times; so, that implies a circumference formula C = 2 X 3.14 X r, which is just = 3.14 X d ... by using 2r is d ("Got it", nod yes. "Yeah!" But, read and think over it again until it really soaks in, if it's not yet crystal clear).

Teacher hint

1. 1
   Help students to really enjoy this exercise. This could be a great turn-on moment, one of those moments where they feel like: "I get it! Wow!", "I like math more than ever". This can be treated like a scientific experiment or even given as an art project (cross teaching), if you're more "artsy" than "math/science buff".
2. 2
   Make-up a mysterious assignment sheet for a class or outside project, if you are a teacher or tutor.
3. 3
   Hint a bit. "Show them, or let them show you, but do not tell them! Let them discover things." If it's a giveaway, then the outcome is too easy for what it is all showing. So instead, make it so that students can discover it as a mystery and have a "Eureka! experience...", not just hear or read about an experiment.
   • You wouldn't want to push straight through a reading or lecture presentation as here, but be subtle at first––lead, facilitate, then clarify it after getting students to present their charts as posters of what they discovered––their way! Students can post their presentations on a math wall, and be proud of their quick-wits, cleverness, working through it!