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This is for the noon day project, and we are putting together two videos.
One will show you how to collect data outside
...how to measure the angle of the sun...
and this video, this tutorial is to show you how to go through the calculations
figuring out the circumference of the Earth.
This is a picture of the Earth and our goal is to figure out how far is to go all the way around one circumference
…and this is replicating the method that Eratosthenes pioneered two thousand years ago.
The first thing I’m going to do is give you a circle because this circle is a model for the Earth
and sometimes we have to keep in mind some of the boundaries and limitations to our model
We are going to use this to represent a cross section of the Earth.
Now if I give you an example here…I’m going to give you a piece of this cross section…
would you be able to figure out how big the entire circumference of this circle originally was?
Now I’ll tell you there are two different methods:
one is to simply use the radius, so 2 x π x r will give you the circumference,
and here we come up against the boundary of this being a model of the Earth,
we can’t dig all the way through the Earth to figure out this radius…
so we’ll try to make use of this distance along the outside of the circle.
Could you figure out the entire circumference given this measurement?
If you think about that for a few minutes and see what you can come up with…
I think this is a simple replica of what we’ll do on a bigger scale…
what you would do is you would take this measure in here probably and multiply it by 4
figuring that this is about one quarter of going all the way around.
Now what you would do is set up an equation such as this
…where the ratio of the distance from A to B over 90 degrees
is equal to the ratio of the distance all the way around, the circumference, over 360 degrees.
so this is 20 centimeters and it would be 80 centimeters all the way around the original circle
very simple, very straight forward,
but it gives you a sense of what we’re gonna try to do to calculate the Earth’s entire circumference
Now if I give you one more difficult example…
this is the same thing except now we don’t have 90 degrees which is very handy and very easy to work with
how’d we again figure out the entire circumference given this piece of pie to work with
and it’s a little bit harder because now we have a degree measure that is not 90 degrees
but we can set up the exact same proportion
we can figure out the angular measurement, I can put the protractor on here,
and it looks like about 37 degrees, and then I can measure this distance here
and again set up the exact same proportion,
and that’s modeling what we’re going to do in the noon day project;
however, again we’re up against another boundary, we have difficulty measuring this interior angle.
Alright, so what Eratosthenes recognized was that if he could measure the angle up here on the Earth,
this representing the Earth’s surface,
if he could measure the angle of the sun here, that angle is the same as the interior angle.
So this angle A is the same as angle B, and you don’t have to dig in the interior Earth to measure,
and then the only other bit of information you need to set up that proportion is the distance along the Earth’s surface…
…so that’s the math that he was using…
now what gets one step even more complicated for the noon day project is this measurement,
this location here, we’re assuming as it is the equator,
and that's not always gonna be the case for what you’re doing.
This looks more like what you’ll be dealing with… your location A or B,
and then you’re going to collaborate with another school from A or B, the other one,
and C as the equator but we don’t have necessarily someone else to collaborate with at the equator.
In this case what we do is instead of just taking the measurement that you would collect
…you’re gonna take that measurement…
if you recognize that this angle here is the same as this angle here…
you’ll recognize that we don’t quite have the interior angle AOB,
we don’t have that angle, we have angle AOC, and that’s no quite it;
but if we can subtract from this angle here, if we can subtract this angle which we can get from location B,
if we subtract that angle from AOC, then we can actually get this interior angle…
So this is the interior angle that we want, and this is the distance that we want,
and then once we have that, that’s what you’re collaborating with other schools to do,
we can set up a proportion: latitudinal distance between A and B
divided by the difference in sun angle between A and B,
so the difference between what was measured here and the angle that's measured here,
it’s gonna be equal to X over 360.
Solve for X and you’ll have the circumference of the Earth.
There are two little assumptions embedded in here and it’s good to revisit with students:
The Earth’s surface is curved. Eratosthenes recognized that and went further exploring to figure out how big it was.
So students don’t necessarily readily accept this or understand how we would know this,
but by figuring out that at the exact same time two locations have different angles for the sun
that would suggest then that the Earth’s surface is curved.
The other assumption is that the rays from the sun are coming in parallel
and that’s because the sun is so large and far away,
so when we go back to this drawing all my lines were drawn exactly parallel
because the sun is so large and far away,
but again it’s an underlying assumption to the calculations we’re doing.
The other tutorial that we will have for you is measuring the sun’s angle outside,
and then you can use that data to collaborate with another school
and try to calculate what the circumference of the Earth is.