Mathematics
UNIVERSITY OF TORONTO
Classroom Adventures in Mathematics – History of Math (23 Mar 2013)
Approximation of Pi
Background information:
he number pi (symbol: π) (/paɪ/) is a mathematical constant that is the ratio of
a circle's circumference to itsdiameter, and is approximately equal to 3.14159. It has been
represented by the Greek letter "π" since the mid-18th century, though it is also
sometimes written as pi. π is an irrational number, which means that it cannot be
expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are
commonly used to approximate π); consequently, its decimal representation never ends
and never settles into a permanent repeating pattern. The digits appear to be randomly
distributed, although no proof of this has yet been discovered.
Activity: Find an Approximate Value For Pi
You will need:
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A piece of card
A compass and pencil
A protractor
A ruler
A pair of scissors
Glue and paper
Step 1
Draw a circle on your card. The exact size doesn't matter, but let's use a radius of 5 cm.
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Mathematics
UNIVERSITY OF TORONTO
Classroom Adventures in Mathematics – History of Math (23 Mar 2013)
Use your protractor to divide the circle up into twelve equal sectors.
What will be the angle for each sector? That's easy – just divide 360° (one complete turn)
by 12:
360° / 12 = 30°
So each of the angles must be 30°
Step 2
Divide just one of the sectors into two equal parts – that's 15° for each sector.
You now have thirteen sectors – number them 1 to 13:
Step 3
Cut out the thirteen sectors using the scissors:
Bahen Centre, 40 St. George St., Room 6290, Toronto, ON M5S 2E4 Canada
Tel: +1 416 978-3323 • Fax: +1 416 978-4107 • www.math.utoronto.ca
Mathematics
UNIVERSITY OF TORONTO
Classroom Adventures in Mathematics – History of Math (23 Mar 2013)
Step4
Rearrange the 13 sectors like this (you can glue them onto a piece of paper):
Now that shape resembles a rectangle:
Step 5
What are the (approximate) height and width of the rectangle?
Its height is the circle's radius: just look at sectors 1 and 13 above. When they were in
the circle they were "radius" high.
Its width (actually one "bumpy" edge), is half of the curved parts along the edge of the
circle ... in other words it is about half the circumference of the original circle. We
know that:
Circumference = 2 × π × radius
And so the width is:
Half the Circumference = π × radius
And so we have (approximately):
Bahen Centre, 40 St. George St., Room 6290, Toronto, ON M5S 2E4 Canada
Tel: +1 416 978-3323 • Fax: +1 416 978-4107 • www.math.utoronto.ca
Mathematics
UNIVERSITY OF TORONTO
Classroom Adventures in Mathematics – History of Math (23 Mar 2013)
With a radius of 5 cm, the rectangle should be:
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5 cm high
about 5π cm wide
Step 6
Measure the actual length of your "rectangle" as accurately as you can using your ruler.
You should be able to measure to the nearest millimetre.
Divide by the radius (5 cm) to get an approximation for π
Put your answer here:
"Rectangle"
Width
Divide by 5 cm
≈π
Remember π is about 3.14159... how good was your answer?
Note: You could probably get a better answer if you:
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used a bigger circle
divided your circle into 25 sectors (23 with an angle of 15° and 2 with an angle of
7.5°).
Selected Curriculum links
Grade 8
http://www.edu.gov.on.ca/eng/curriculum/elementary/math18curr.pdf
“Measurement: converting between cubic centimetres and cubic metres and between
millilitres and cubic centimetres; developing circumference and area relationships for a
circle; developing and applying the formula for the volume of a cylinder; determining
and applying surface-area relationships for cylinders”
Grade 12 college preparation course Mathematics for College Technology
“Students apply geometric relationships to solve problems involving composite shapes
and figures and investigate the properties of circles and their applications”.
Bahen Centre, 40 St. George St., Room 6290, Toronto, ON M5S 2E4 Canada
Tel: +1 416 978-3323 • Fax: +1 416 978-4107 • www.math.utoronto.ca
Mathematics
UNIVERSITY OF TORONTO
Classroom Adventures in Mathematics – History of Math (23 Mar 2013)
Resources:
• Estimation of pi
o http://www.mathsisfun.com/activity/pi-approximation.html
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What is pi and some historical information about pi o http://en.wikipedia.org/wiki/Pi links to other interesting information and activities involving pi o http://www.joyofpi.com/pilinks.html Bahen Centre, 40 St. George St., Room 6290, Toronto, ON M5S 2E4 Canada
Tel: +1 416 978-3323 • Fax: +1 416 978-4107 • www.math.utoronto.ca