Discover Pi for Yourself Using Circles
How was the math constant called "pi" discovered -- and could you have discovered it? Well, yes, with a bit of close work, you can uncover the clever idea and source of the concept, as well as get its no-longer abstract meaning and find an approximate value. It is wrapped up in every circle and sphere––but where and how could you have envisioned it in the nature of circles? Keep reading for detailed instructions for your jump into discoveries in math.
Contents
Steps
Basic geometry of the circle in a plane
- 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<b> - is short for "of equal distance"
- <b>circle - all the points equidistant, from the center (center point).
- 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
- 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.
- 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
- Number four different sizes of circular containers or lids. A globe or ball (sphere) can work also, but it's harder to measure.
- Get a non-stretchy, non-kinky string and a meter-stick, yardstick or ruler.
- Make a chart (or table) like the following one:
Circumference | diameter | quotient C / d = ?- __________|________|__________________
- __________|________|__________________
- __________|________|__________________
- __________|________|__________________
- 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.
- 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. - 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).
- 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".
- 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.
- 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).
- Finally, take the diameter string and use it to cut its length off the circumference string three times. Do this for each of the containers. The left-over piece of string from each of the circumference strings cut-outs will be approximately the same length. The measurement length of this short piece of string should be .1415 which is just an example of getting approximately 3.14...
Teacher hint
- 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/more than I thought". Treat this like a scientific experiment, as sort of a "math/science" cross-curricular assignment.
- Make-up a mysterious assignment sheet for a class or outside project, if you are a teacher or tutor.
- 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!
- Use this as a great in-class project (cross teaching) "art-math-art" assignment––or for your students to take home as a project for extra credit outside math class. And, after you apply this one, you might like to explore leading to be a great teacher.
Tips
- (By the way: the arc on a circle which is as long as the radius is called a "rad." It is a constant used in trigonometry and calculus.)
- That little fraction more than 3 times that the diameter will fit around the circle is about 1/7 of diameter = approximately 0.14, and 3 X (7/7) = 21/7 and that plus the 1/7 is 22/7 = 3.14 approx, but the larger the circle the more that inaccuracy will be apparent (0.14 X 7 = 0.98, off by 0.02 = 2/100 = 2% under diameter; actually 22/7 is more accurate than 3.14, but this value 22/7 is about 1/8 of 1% of diameter over valued).
- Formula: Circumference = pi X diameter.
- Solve for pi as follows:
- Solve for pi as follows:
C = pi X d
C/d = (pi X d)/d
C/d = (pi)d/d
C/d = pi X 1 because d/d = 1 so that gives us
C/d = pi
The ratio C/d "defines" the constant pi, regardless of a circle's size, in geometric equations, but π also occurs in areas of mathematics that do not directly involve geometry.
- You can see historical listings on a chart for value of pi and their chronology/timeline, showing early ideas on through modern calculations of millions of digits.^{[1]}
- Pi is the letter p, π in Greek. A stated approximation of pi was devised by the Greek philosopher Archimedes of Syracuse (287-212 BC). He obtained the following inequality:
223/71 < π < 22/7
Archimedes knew that π does not equal 22/7, but made no claim to have discovered a more exact value. If we estimate pi as the average of 223/71 and 22/7, then his two bound give us 3.1418, an error of about 0.0002 (two 100ths of 1% error).^{[2]}- About fifteen centuries earlier than Archimedes the Egyptian Rhind Mathematical Papyrus, a page from an ancient text explaining math problems, used "pi = 256 / 81". That is (16/9)^{2}, about 3.16 (compare that to 25/8 = 3.125).^{[3]}
- Archimedes (around 250BC) also used value of pi = 256/81 = sum of = 3 + 1/9 + 1/27 + 1/81, and also the Egyptians using 3 + 1/13 + 1/17 + 1/160 (= 3.1415) for pi in problem 50 of the Egyptian Rhind Mathematical Papyrus.^{[4]}
Things You'll Need
- 5 different sizes of circular containers (small, medium, large, larger, or very large)
- String (not stiff or kinky)
- Tape/pins
- Meter-stick, yardstick or ruler
- Chart
- Pen or pencil
- Calculator (optional if you need one)
Related Articles
- How to Lead to Be a Great Teacher
- Center a Circle
- Prove That Pi Is Not 3
- Falsely Prove That Pi Equals 3
- Type the Pi Symbol
- Calculate the Circumference of a Circle
- Calculate Pi by Throwing Frozen Hot Dogs
Sources and Citations
- history HistTopics Pi_chronology http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi_chronology.html
- "Pi through the ages" http://www.gap-system.org/~history/HistTopics/Pi_through_the_ages.html
- "Rhind Mathematical Papyrus"
- University of Buffalo, New State Univ system, "Determining the Value of Pi" -- http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_geometry.html
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