AB Calculus Summer Exploration #2
Archimedes and Pi
Name:___________________
The Greek mathematician, Archimedes, is considered by many
to be the greatest mathematician of antiquity. He was born in
287 BC in Syracuse, Sicily. In addition to being a great
mathematician, he was a prolific scientist, engineer, and
inventor. He is famous for his work in hydrostatics and for his
inventions of war. He created siege engines, proposed the use
of levers to lift ships from the sea, and invented the screw
which bears his name. But if asked, he would surely say that
pure mathematics was his true love. He was killed by a Roman
soldier, and legend has it that his last words were "Do not
disturb my circles"
http://t2.gstatic.com/images?q=tbn:LdBSjasGIO31tM:http://www.fredsakademiet.dk/library/science/_gifs/archimedes.jpg
In this exploration, we will use Archimedes “method of reduction” to approximate the value of pi
to a high level of accuracy, investigating the limits within which the answer lies. We will look at
the perimeters of polygons of increasing number of sides – both inscribed and circumscribed
about a circle. We will use our work to find an approximation of pi.
In the “Figure” column of the table, draw an n-sided regular polygon inscribed in the given circle
of radius 1 cm. Let one base be parallel to the bottom of your paper. Draw another n-sided
regular polygon circumscribed about the given circle, and concentric with the inscribed polygon.
Use a ruler and draw neatly. See pentagon example below.
r=1 cm.
Inscribed
n-gon:
Circumscribed
n-gon:
π/n
π/n
2π/n
1 cm
bi/2
n = 5 (pentagon)
1 cm
bc/2
One can compute each polygon’s perimeter separately, but that would be inefficient and could
lead to error. We will go about generalizing the process as follows. Divide the n-gon into n
isosceles triangles (one is shaded above) which share the circle’s center as a vertex so that the
vertex angle is 2π n . (They should look like pie wedges – pun intended.) The total angle
measure for all n central angles must be 2 pi radians. Yes, we will be working in radians in
calculus. Consider the triangle with its base parallel to the bottom of the paper. If we drop an
altitude from its vertex, we get the resulting right triangles to the right, and we can write an
equation for each base as a function of angle π n .
1
Using SOHCAHTOA, write an equation below for sin
π
n
using the inscribed base and solve for bi.
Then show that the base of the inscribed n-sided regular polygon, bi, is bi = 2 sin
Using SOHCAHTOA, write an equation below for tan
π
n
FG π IJ .
H nK
using the circumscribed base and solve
for bc . Then show that the base b of the circumscribed n-sided polygon is bc = 2 tan
Since there are n bases, the perimeter of the inscribed polygon is Pi = n • 2 sin
Similarly, the perimeter of the circumscribed polygon is Pc = n • 2 tan
FG π IJ .
H nK
FG π IJ .
H nK
FG π IJ .
H nK
We know that the ratio of the circumference of a circle to its diameter is pi or
πd
d
= π . If we
divide each perimeter by the 2 cm diameter of the circle, we will find an approximation of pi.
Inscribed:
FG IJ
H K
Pi
π
= n sin
d
n
Circumscribed:
FG IJ
H K
Pc
π
= n tan
d
n
Use the equations to complete the table “Perimeter Exploration” for each n-gon. Rather than
compute each cell individually, you can graph the equation for the inscribed ratio Pi /d as y1 and
the circumscribed Pc /d as y2 with your calculator. You will need to use x in place of n. (If you
have a TI89, you will need to put the multiplication symbol between x and the trigonometric
function.) Check that you are in radian mode! Setup your calculator table with TblStart 4 for
the 4-sided polygon, and let ∆Table be 1 to get each subsequent n-gon. You can also find the
average of the inscribed and circumscribed P/d with your calculator by entering
y 3 = ( y1 + y 2) / 2 . You must type y1(x), etc. if using a TI89. Use VARS key to find Y1 and Y2 if
using TI83.
Complete the table on the next page, showing answers with as much accuracy as your calculator
will display in the table. Then answer the questions that follow the table on a separate piece of
paper.
2
Perimeter Exploration:
Figure of
Number
n-gon
of sides,
n
Inscribed
Pi /d
y1
Circumscr.
Pc /d
y2
Average
P/d
y3
4
5
6
8
12
24
48
96
Archimedes used 6, 12, 24, 48, and 96-sided polygons. See how far you get with drawing the
inscribed and circumscribed polygons. At some point, there will be too many sides to draw!
3
On a separate piece of paper, write out complete responses to the following questions about the
perimeter exploration:
1) Do the inscribed and circumscribed P/d values appear to be getting closer to one another for
higher x values? Why does this make sense?
2) Scroll down your table until you find an x value for which the displayed y1 and y2 values are
the same. How many sides does the polygon have that corresponds to that y-value?
3) What do you notice about that y value?
4) How does your best approximation of pi compare to Archimedes’ approximation, as excerpted
from Proposition 3 of his treatise Measurement of a Circle:
The ratio of the circumference of any circle to its diameter is less than 31/7 but greater than 310/71.
(Hint: Convert Archimedes mixed fractions to decimals for comparison.)
Repeat the exploration for areas of the inscribed and circumscribed polygons. Develop your own
equations for the area of individual isosceles triangular pie wedges, and then find the sum of all
the wedges. For the inscribed n-gon area, you may want to use A =
1
ab sin C for the area of a
2
triangular wedge, where a and b are lengths of 2 sides, and C is the included angle. For the
circumscribed n-gon, you will want to use the traditional formula for the area of a triangular
wedge, A =
1
bc h . Remember, you already know a trigonometric expression for bc . Show all
2
work for your area equations below. Be neat and organized! Use your calculator to complete the
“Area Exploration” table on the next page. Show answers with as much accuracy as your
calculator will display in the table. Answer the same four questions above as they pertain to
area.
4
Area Exploration:
Number
of sides,
n
Figure of
n-gon
Inscribed
Area Ai
4
5
6
8
12
24
48
96
5
Circumscr.
Area Ac
Average
Area