Lab
Indirect Measures: Inscribed Circles
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In this lab you are going to estimate the value of
based on probability ratios. You will be given a target paper with five circles inscribed within five squares. Using a centimeter ruler you are first to measure the length of each square and record your answers to two-decimal places in the data table below.
Next completely cover your target paper with a piece of carbon paper, placing it inky-side down. Now drop a marble 100 times onto the carbon paper. Make sure that you randomly release the marble so that each part of the paper receives relatively the same number of impacts.
When you are finished, count the number of hits that are totally within each circle. Do NOT count a hit that lies on the edge of a circle. Next count the total number of hits that lie completely within each square - including the hits on the edge of the circle, within the circle, and in the area between the circle and the square. Do NOT count the hits that fall on the edge of the square. Record your answers in the data table below.
The left picture represents the hits landing on one circle/square. On the right picture, the dots which need to be discarded have been marked. Dots circled on the edge of the circle count for the square but not for the circle. Dots on the end of square do not count anywhere. Obviously, any dots that fell outside of the square were not counted.
number of dots in the circle = 39
number of dots in the square = 39 + 15 = 54
the ratio of hits would be: 39/54 = 0.722 or 72.2%
Data table:
square being
studied
square's side
length
hits within circle
hits within square
ratio of hits circle/square
(cm)
(decimal)
#1
#2
#3
#4
#5
Analysis:
Now we will use our results to calculate an experimental value for
. For each circle, the ratio of the number of "circle hits" to the number of "square hits" represents an experimental percentage of how much of the square's area is covered by its inscribed circle. Recall in our example that the ratio was 39/54 = 0.7226 or 72.2%
Remember from geometry that the area of a square is given by the formula
, while the area of a circle is given by the formula
. Since each circle is inscribed within its respective square, the diameter of each circle equals the length of the side of its square. So our formula becomes
. Having measured the side of each square,
s
, we can now determine the area of each incribed circle.
In our example:
s
_{square}
= 5.84 cm
s
^{2}
_{square}
= 34.1 cm
^{2}
Using our relationship that the area of an inscribed circle = (ratio of hits)(area of its square), we determine that the numerical value of our sample circle's area is: A
_{circle}
= 0.722(34.1) = 24.6 cm
^{2}
.
Now we can use the formula
to calculate an experimental value of
. Substituting in the values from our example we would get
24.6 = ¼
(34.1)
= 2.89
Error Calculations:
To determine how closely our results come to
, we will calculate a percent error against each circle's result. Since
is a transcendental number, it does not have a limited number of significant figures. However, your measurements were limited to two-decimal places so we will use 3.14 as our accepted value for
. The formula for percent error is
For our sample circle/square, our error would have been 7.96%.
square being
studied
experimental
value
% error
#1
#2
#3
#4
#5
Conclusions:
Which of your circles had the lowest percent error?
1
2
3
4
5
Why do you think that this circle was the best? Was it the circle's location on the paper? the size of the circle? that it received the greatest number of hits? Explain and elaborate.
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