1 / 168

Measuring, constructing, and using angles

Measuring, constructing, and using angles . Contents Why Measure Angles? How might we measure angles? What is an “angle”? What does “measurement” mean? Our “measuring tool” for angles. Contents (continued) Measuring an angle Drawing an angle of a given size

bridie
Download Presentation

Measuring, constructing, and using angles

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Measuring, constructing,andusing angles

  2. Contents Why Measure Angles? How might we measure angles? What is an “angle”? What does “measurement” mean? Our “measuring tool” for angles.

  3. Contents (continued) Measuring an angle Drawing an angle of a given size The “radian” unit of measure for angles.

  4. Why measure angles?

  5. Why measure angles? So we can figure outhow big the Earth is!

  6. If we choose two places that are north and south of each other, A B

  7. then measure the distance between them, A Distance B

  8. and also the highest angle the Sun reaches at each place on a given day, Highest angle at Location A Highest angle at Location B Sky simulations made with the free astronomy program Stellarium (http://stellarium.org/)

  9. we can calculate the size of the Earth. The first person to do this calculation was Eratosthenes, a librarian at the Great Library of Alexandria, Egypt, around 200 BC.

  10. Actually, almost all of astronomy and geography depends upon being able to measure angles. So do many jobs, such as surveying.

  11. So, how might we measure angles? First, let’s review what we mean by “angle”, and “measurement”.

  12. One definition of “angle” is“A pair of rays thathave the same endpoint”. The endpoint, V, is called the vertex.

  13. You’ve probably learned that angles are related to “turns”, and that the same set of rays can be made by two different turns. For example, here are …

  14. the two “turns” that take us from ray VA to ray VB. Counterclockwise turn Clockwise turn

  15. Now that we’ve reviewed what “angle” means, what do we mean by “measurement”?

  16. A good definition is found in The Archimedes Codex: How a Medieval Prayer Book is Revealing the True Genius of Antiquity’s Greatest Scientist:* * RevielNetz and William Noel, Da Capo Press, 2009, p. 41.

  17. According to the authors, “To measure is to find a measuring tool and apply it successively to the object being measured. Suppose we want to measure a straight line.

  18. “For instance, suppose we want to measure your height, which is really saying that we want to measure the straight line from the floor to the top of your head.

  19. “Then what we do is take a line the length of an inch [this is our measuring tool] and apply it successively, well over sixty times, but probably fewer than eighty times to measure your height.

  20. “Since this is very tiresome, we have pre-marked measuring tapes that save us the trouble of actually applying the [one-inch line] successively,

  21. “but, at the conceptual level, successive application is precisely what takes place.”

  22. That definition of “measurement” needs some explanation. For example, when we read “To measure is to find a measuring tool and apply it successively to the object being measured,”

  23. we probably thought of “measuring tools” as rulers, etc. Actually, the authors meant something quite different.

  24. They meant that to measure a length (that’s the example they give), the “tool” we choose is a line segment with a convenient length.

  25. The authors mentioned “an inch” as an example of a convenient length, but we could use a segment of any length we like.

  26. For example, we could use this yellow segment:

  27. Now, suppose that we wanted to “measure” this red segment.

  28. We’d start at one end of the red segment, and “apply” the yellow segment to it repeatedly, until we got to the other end …

  29. 1

  30. 2

  31. 3

  32. Here’s the summary of what we just did to “measure” the red segment. 1 2 3

  33. Because we had to apply the yellow segment three times, 1 2 3

  34. we say that the length of the red segment is “3 yellow segments”. 1 2 3

  35. For example, if our yellow segment were an inch, the length of the red one would be “three inches”. 1 2 3

  36. As the authors of The Archimedes Codex told us, this process would be tiresome if we had do many measurements.

  37. For that reason, we make pre-marked rulers and measuring tapes, 2 4 5 1 3

  38. so that we may “apply” several “yellows” at once. 2 4 5 1 3

  39. If we wished, we could also mark off fractional parts of our yellow segment, just as we do with inches on a ruler.

  40. Now that we’ve reviewed what “angle” and “measuring” mean,

  41. let’s learn the common ways of measuring an angle.

  42. As we’ve seen, the “measuring tool” that we use for lengths is some convenient segment.

  43. However, when we measure angles, the “measuring tool” that we use is some sector of a circle.

  44. For example, this one:

  45. To measure an angle,

  46. we align one side of our sector with one of the angle’s rays, Note: “Point” of the sector must be aligned with vertex of the angle.

  47. then keep “applying” our sector until we reach the other ray. Note: “Point” of the sector must be aligned with vertex of the angle.

  48. then keep “applying” our sector until we reach the other ray. Note: “Point” of the sector must be aligned with vertex of the angle.

  49. We use the same method for the other direction of “turn”:

  50. Note: “Point” of the sector must be aligned with vertex of the angle.

More Related