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Circumference and More Geometry - Contenido educativo

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Subido el 28 de mayo de 2007 por EducaMadrid

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NASA Connect Segment explaining questions about Erastothenes, the Earth's circumference, parallel lines, angle relationships, and a transversal.

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Did you know that over 2,000 years ago a Greek librarian used geometry to determine the circumference of the Earth? 00:00:00
I did aerostomes and used to find the circumference of the Earth. 00:00:12
What are the angle relationships between parallel lines and a transversal? 00:00:15
The concept of the Earth being a large sphere was not unknown to the ancient Greeks. 00:00:21
An everyday observation, such as the disappearance of ships below the horizon, 00:00:26
indicated that the Earth might be spherical or round. 00:00:30
But how large was it? 00:00:33
The person who figured it out was a librarian named Aristophanes, 00:00:35
who lived in Alexandria, Egypt, about 300 BC. 00:00:39
While looking through a scroll one day, he read that at noon on the longest day of the year, 00:00:43
a vertical column cast no shadow in Cyene, a city south of Alexandria. 00:00:48
Aristophanes knew that this did not happen in Alexandria. 00:00:53
He thought to himself, 00:00:56
how was it possible to have shadows in Alexandria and not in Cyene at the same time of day? 00:00:58
Aristophanes figured out that the sun must be directly overhead in Cyene, but not in Alexandria. 00:01:03
Aha! Here was proof that the Earth's surface is curved. 00:01:10
Using a little geometry, Aristophanes set out to determine the circumference of the Earth 00:01:15
and find out just how big it is. 00:01:20
Just like our pizza example, 00:01:22
if our friend Aristophanes could determine the central angle at the center of the Earth 00:01:24
and the length of the edge or arc, 00:01:30
then he could figure out the circumference of the Earth. 00:01:33
Now, finding the length of the edge or arc was fairly simple math. 00:01:36
Aristophanes asked a friend to walk from Alexandria to Cyene 00:01:41
to measure the distance between the two cities. 00:01:46
His friend estimated the distance to be around 800 kilometers or about 500 miles. 00:01:49
Finding the central angle, however, would take some geometry. 00:01:56
First, Aristophanes assumed correctly, I might add, 00:02:00
that the sun's rays are parallel since the sun is so far away. 00:02:04
Check this out. 00:02:08
In this diagram, we can see that there is no shadow at Cyene, 00:02:09
while there is a shadow in Alexandria. 00:02:12
The line that is formed by the gnomon, or vertical column at Alexandria and the center of the Earth, 00:02:15
cuts or intersects the two parallel lines formed from the sun's rays. 00:02:21
A line that intersects two parallel lines is called a transversal. 00:02:27
The two angles formed from the transversal line and the parallel lines 00:02:31
are called alternate interior angles. 00:02:35
And, according to geometric rule, they are equal. 00:02:38
Let's prove it. 00:02:42
Take a piece of paper of any width and draw a diagonal line on it. 00:02:43
Label the angles A and B just like this. 00:02:48
Now, cut the paper along the diagonal so you have two triangles. 00:02:51
Compare angles A and B by placing one angle on top of the other. 00:02:56
Hey, what do you notice? 00:03:00
The angles are equal no matter what size paper you started with. 00:03:02
Right. 00:03:05
When two parallel lines are intersected by a transversal, 00:03:06
the alternate interior angles are equal. 00:03:10
Huh, Aristophanes was quite a geometer. 00:03:14
From his measurements, Aristophanes calculated the sun's rays made an angle of 7.5 degrees at Alexandria. 00:03:17
Now, since this angle was formed by two parallel lines and a transversal, 00:03:24
the central angle of the Earth must also be 7.5 degrees. 00:03:29
By knowing these two things, the central angle and the distance from Alexandria to Syene, 00:03:33
Aristophanes calculated the circumference of the Earth. 00:03:39
360 degrees divided by 7.5 degrees equals 48 slices of the Earth. 00:03:42
Are you still with me? 00:03:51
Okay, hang tight. We're almost there. 00:03:53
Now, if you remember that the estimated distance between Alexandria and Syene is 800 kilometers, 00:03:54
and you multiply that distance by the number of slices in the Earth, 48, 00:04:00
what is the circumference of the Earth? 00:04:06
Well, if you estimated that distance to be 38,000 kilometers, you're absolutely right. 00:04:10
Aristophanes' estimate was really close to the Earth's circumference, which is 40,074 kilometers. 00:04:17
His percentage error was about 5%, and was probably due to an error in the distance between the two cities. 00:04:25
5%? Huh, that's pretty good, 00:04:32
considering Aristophanes used only his feet, his eyes, his imagination, and, of course, his knowledge of geometry. 00:04:35
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Idioma/s:
en
Materias:
Matemáticas
Niveles educativos:
▼ Mostrar / ocultar niveles
      • Nivel Intermedio
Autor/es:
NASA LaRC Office of Education
Subido por:
EducaMadrid
Licencia:
Reconocimiento - No comercial - Sin obra derivada
Visualizaciones:
433
Fecha:
28 de mayo de 2007 - 16:51
Visibilidad:
Público
Enlace Relacionado:
NASAs center for distance learning
Relación de aspecto:
4:3 Hasta 2009 fue el estándar utilizado en la televisión PAL; muchas pantallas de ordenador y televisores usan este estándar, erróneamente llamado cuadrado, cuando en la realidad es rectangular o wide.
Resolución:
480x360 píxeles
Tamaño:
28.35 MBytes

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