Eratosthenes
Purpose
Students will find
the circumference of a wheel (or globe) directly and indirectly, and compare
their results.
Students will prove
if the claims of Eratostehenes are true.
Materials
Wheel (or small globe or ball)
Ruler
Protractor
Paper and pencil
Directions
1. First Method:
a. Placing the wheel
(or globe) with the center on the paper near one edge, draw the
outline of the circle
on the page, marking the center.
b. Draw a radius line
from the center mark to the outside of the circle and ~3 cm
beyond. This line
represents a building standing perpendicular to the horizon at that
point on the circle.
c. Draw a second
radius line from the center perpendicular to the first radius.
This line can
represent a well that is perpendicular to the horizon at another point on
the circle.
e. Use the ruler to
measure the distance (s) along the circle between the two lines.
f. Measure the
circumference (TV). Then compare to the C using the formula below.
C = 360°/(s*90)
2. Second Method:
a. Use the ruler to
measure the diameter (d) of the wheel (or globe).
b. Use the formula C=Π∗d to
calculate the circumference of the wheel.
3. Third Method: Use
a tape measure wrapped around the wheel (or globe) to find the circumference.
4. Using any measuring device, measure the height of
one of your group mate (TV). Measure the shadow casted by your group mate (x),
then calculate for the height (h) which will be your observed value.
h
= x/sin ɵ
Where ɵ is: 30 if 1 pm or 11 am
60 If 2 pm or 10 am
90 If 3 pm or 9 am
120 If 4pm or 8 am
150 If 5 pm or 7 am
Table
2.1 Measured Dimensions of a circle
|
Circular
Object
|
A
(angle)
|
S
(side)
|
D
(diameter)
|
|
|
|
|
|
Table
2.2 Comparative Data of The Calculated Circumference Using Different Methods
|
Method
Used
|
Circumference
|
%
Error
|
|
First
|
|
|
|
Second
|
|
|
|
Third
|
|
|
|
|
|
|
Table
2.3 Comparison of Actual Height Using Shadow
|
Time
|
Actual
height (TV)
|
Shadow
Height (x)
|
Calculated
Height (h)
|
%
Error
|
|
|
|
|
|
|
Guide
Questions
1. Compare your
values for all three measurements/calculations.
2. Using the result
of your third method as the true value, find the % error of each method.
3. What is the claim
of Eratosthenes? Why would Eratosthenes' method not work if the earth were
flat?
4. Why is it
important that we assume that the Sun's rays arrive in parallel on the Earth?
5. How would your
calculated circumference change if you measured the angle larger than it really
is?
6. How would your
calculated circumference change if you measured the angle smaller than it really
is?
7. What could account for errors in your value?
Evaluate your methods and brainstorm ideas for
improving your
results.
Rlp2009
