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## NASA

### Course: NASA > Unit 2

Lesson 2: Measuring the solar system- A flat earth
- Arc length
- Circumference of Earth
- Occultations
- Occultation vs. transit vs. eclipse
- Size of the moon
- Angular measure 1
- Angular measure 1
- Trigonometric ratios in right triangles
- Angular Measure 2
- Angular Measure 2
- Intro to parallax
- Parallax: distance
- Parallax method
- Solar distance
- Solve similar triangles (advanced)
- Size of the sun
- Scale of solar system

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# Circumference of Earth

Created by NASA.

## Want to join the conversation?

- Where was Eratosthenes born?(11 votes)
- He was born in Cyrene in modern day Libya.(12 votes)

- How did Eratosthenes send message from Syene to Alexandria about the solstice occurring in Syene to measure the shadow of the pole present at Alexandria?(6 votes)
- I think he knew there was no angle at summersolstice, so he only had to measure the length of the shadow at that particular moment at 12 o'clock(6 votes)

- How did Eratosthenes check on his pole at the same time as the other person checked his at noon i.e. how did Eratosthenes know that it was noon at Syene and that he should check his pole's angle?

Is it because of knowing the time of travel from Alexandria to Syene?(4 votes)- Alexandria and Syene were close enough together (they're both in Egypt) that they used the same time. Essentially, they're in the same time zone. Also Syene is almost directly South of Alexandria meaning that they hit their solar noon (i.e. the actual time when the Sun is directly overhead) just about at the same time.

I've actually lead this experiment with my students before. The important consideration is knowing the time of your local**solar**noon, which is not necessarily12:00p.m. exactly. On any given day, one point on your longitude (North-South axis) will be directly under the sun at the solar noon: the same position as Syene in the classical problem. If you measure your distance to that point, and measure your shadow at the solar noon, you can calculate the circumference of the earth. Of course, today we have modern tools that can tell you all this information, but it's not impossible to conceive how Eratosthenes gathered this information more slowly and scheduled his measurements at the right time in his age.(3 votes)

- Ok that gives us circumference using that equation based on assumptions. . but where can we do a hard and fast scientific experiment and actually measure the curvature. . and can I assume that the rate would be close to. . 1 mile=8 inches ×the number of miles squared? And if that is correct could you please put it into equation form. . and I'm getting nothing but straight and flat with a telescope at over 100 miles. . so how and where could I do it that would show that ratio of curvature(3 votes)
- Can someone answer this question please?(1 vote)

- Hi thanks for the information.

I have a question I was wondering if you could help me on. It seems this whole calculation is dependent on the assumption that the sun rays are arriving to Earth in a parallel manner. At the time of Eratosthenes what proof is there that this was correct? Cheers.(1 vote)- 24,901.55/360 =69.1709722 x 16.39721 = 1134.21095

pi x 360 = 1131.408/69 = 16.39721

16.39721 x 22.00 = 352 + 8 = 360

intercalculate from a few decimals and same numbers occur

3.1428 X 360 = 1131.408

24,901.55/1131.408 =22.009345877(4 votes)

- But how can someone not living on the equator measure the circumference ??(1 vote)
- Neither syene nor alexandria are on the equator. Being on the equator or not is irrelevant information.(3 votes)

- How far is it from Alexandria to Syene, the two cities mentioned in the video(1 vote)
- or 500 miles. He states this in the video I believe.(2 votes)

- If the circumference wasnt known till modern day, how would they figure it out?(1 vote)
- You mean, other than the method just described in this video?

Eratosthenes used this method to calculate the circumference of the Earth, a method that assumed the Earth is Spherical and used the mathematical properties of spheres and circles with basic angle math to calculate the circumference of the Earth, much like you can calculate the circumference of any circle.

Modern day tools have refined our measurement of the Earth's circumference and allowed us to confirm/compare with Eratosthenes result.(2 votes)

- Can a city with different latitude where the angle is measured be at a different longitude? If it is at different longitude, can we just measure the vertical distance between the corresponding latitudes and measure angles at the local noon time for this method to work?(1 vote)
- Were there any smaller units of stade? Like today, Meter is cm , and that is mm(1 vote)
- In terms of the egyptian stade there where also khet and cubit(1 vote)

## Video transcript

[MUSIC PLAYING] In this video, we're going
to talk about Eratosthenes, who was a Greek scholar that
lived about 2,000 years ago. Eratosthenes found a way,
using none of the modern tools that we have, to measure the
circumference of the Earth. And in this video, we're
going to see how he did this. So the heart of
Eratosthenes's measurement is a simple geometry problem. So consider the
circle shown here, which has points A
and B. And let's say that we know the distance
that A and B make on the circumference
of the circle. So we know the
measure of arc AB. Now, the question is,
with this knowledge, can we determine the
circumference of our circle? And I'm sure you all are
thinking that the answer is obviously no because
A and B are just two random points on the circle. So just knowing their distance
doesn't help us very much. What we need is some
information that makes A and B not just random
points on the circle anymore. We need to know the
angle that A and B make with the center of the circle. Once we know this,
we know how far around A and B go on
the circle because we know that every circle has
in it 360 degrees in one full revolution. So by knowing theta, we know
the fraction of the circle that the arc AB takes up,
and we can simply extrapolate to find the circumference. So let's make this
one a little clearer with some concrete examples. So let's look at the
circle on the right now. And again, we have
points A and B. And here you can clearly
see that the angle that A and B make with
the center of the circle is 90 degrees. And since we know that 90
goes into 360 four times, arc AB is 1/4 around
the circle, meaning that the piece shown
here, the shaded piece, will fit into the
circle four times. So in this case, the
circumference of the circle is four times the
length of arc AB. Now let's just
drive the point home further with this other circle. Again, we have points
A and B. And here the angle between points A
and B, let's say we measure, and it turns out
to be 36 degrees. So since 36 goes
into 360 ten times, we know that 10 of these pieces
will fit into the circle. And in this case,
the circumference is 10, which is the
number of pieces, times the length
of one piece, AB. Now, in general, the
circumference of the circle is given by the number
of pieces we have times the length of one piece. And just right
inside explicitly, the number of pieces
is 360 degrees divided by theta, the angle
that A and B make with the center of the circle. And the length of one
piece is simply AB. So now, if we just know
these two things, the arc AB and the angle A and B make
with the center of the circle, we can determine
the circumference. And this is really the heart
of Eratosthenes's method. So now let's apply this. Now we have another circle,
but this time we'll explicitly identify the circle as
the Earth, and point A becomes a city, Alexandria. This is a city in
Egypt, and this is where Eratosthenes lived. Point B now becomes
the city of Syene. So in Eratosthenes's
day, it was known that the distance between
Alexandria and Syene was about 500 miles. Of course, back then
the units weren't miles. But we know the
conversion factor, so we don't have to worry
about the old system of units. The distance between Alexandria
and Syene is 500 miles. And now, looking back
at our previous problem, we see that all we have
to do to figure out the circumference of the Earth
given this information is to figure out the angle that
Alexandria and Syene make with the center of the Earth. And the really brilliant thing
about Eratosthenes's method is that he found a nice
way to measure this angle. So how did he do this? It was known that in Syene there
was a well, a long, deep well such that at noon on
the summer solstice you could see the
sun's rays light up in the bottom of the well. And if you think about
this, what this means is that since this well
is such a deep thing, this means that
the sun's rays must have been coming into the
Earth parallel to the well. So we draw these rays, and
we see the sun is very, very far away from the Earth. And because the
sun is so far away, we can treat the rays
coming in from the sun at different points as parallel. So here we've drawn the ray
that comes in at Alexandria and the ray at Syene. Now, it's an interesting
property of parallel lines that if we have these two
parallel lines shown here, the two rays, and we have
the radii also shown, this angle that we've called
theta 2 is equal to the angle that we're trying
to find, theta 1. This is a fact that you
might have been exposed to in your geometry classes. It's called the property
of corresponding angles. But regardless, it's pretty
easy to prove to yourself. So since we know this, now we
can transform the hard problem of measuring theta 1 to the
relatively easier problem of measuring theta 2. So how do we measure theta 2? It's pretty straightforward. So let's zoom in on this region. We have the surface of the Earth
as one of the important lines. And then the radius
gets translated into a vertical stick. And we also have the sun
with one of its rays. So this ray casts a
shadow on the ground. And by knowing the
length of the shadow and the height of our stick,
we can construct this triangle and simply measure
the angle theta 2. All we have to do is look at the
shadow created off of a stick. And this is exactly
what Eratosthenes did. And he measured that
the angle theta 2 was equal to 7.2 degrees. So now we have this information. The angle theta, which
is called theta now, is equal to 7.2 degrees. And the measure of the arc
AB, or Alexandria to Syene, is 500 miles. So with this information,
now all we have to do is feed this into the
formula that we got earlier. Let's just recall it--
C equals 360 degrees divided by theta times AB. And so this is
very simple to do. We plug in 7.2
degrees for theta. We plug in 500 miles for AB. And in the end, we find that C
is equal to 50 times 500 miles, or 25,000 miles. So this is our estimate,
or Eratosthenes's estimate, for the circumference
of the Earth. Let's call that C Eratosthenes. And so see how simple it
was for us to get this. But notice that, according
to our modern measurements, the average circumference
of the Earth, because it's not a perfect
sphere, is around 24,900 miles. So we were only
about 100 miles off with this seemingly primitive
method, which is about half a percent off. So this is a very
impressive thing for someone who lived so long ago without
the access to these tools that we now use. [MUSIC PLAYING]