Unit Circle: 360 Degrees = 2(pi) Radians

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The discussion centers on the relationship between degrees and radians in the context of the unit circle, emphasizing that 360 degrees equals 2(pi) radians regardless of the circle's radius. It clarifies that while the arc length is proportional to the radius, the angle measure in radians remains constant for a full rotation. Participants explore the definition of a radian, noting it as the arc length divided by the radius, and highlight that for any circle, the angle corresponding to a full rotation is always 2(pi) radians. The conversation also touches on the concept of radians being a distinct unit of measurement compared to degrees, yet both measure angular rotation. Ultimately, the key takeaway is that the relationship between degrees and radians is consistent across circles of different sizes.
Miike012
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The book talks about a unit circle...
360 deg = 2(pi) rad

if it wasnt a unit circle... say r = 4
would it then be
360 deg = 8(pi) rad?
 
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Imagine a circle with a radius length of 1 unit. Form a segment, also of unit length, intersecting a point on the circle and with the other endpoint at the center of the circle. ROTATE this segment keeping the centerpoint of the circle the same so that the rotated segment now intersects a different point on the circle. The length of the arc swept by this rotation is the number of radians of the rotation.

You should also know that often the number of radians of an angle is given in terms of pi.
 
if it wasnt a unit circle... say r = 4
would it then be
360 deg = 8(pi) rad?
 
Nevermind... it would still be 180 deg = 2(pi) rad no matter the radius... Right?
 
Miike012 said:
Nevermind... it would still be 180 deg = 2(pi) rad no matter the radius... Right?

No. .
 
Then if it was 4 = r
it would be 8(pi) rad = 360 deg?
 
Also.. what is the point of telling us that the radius equals the arc? This isn't exactly how it is stated but I think you know what I am talking about.. if not... ill find the exact wording.
 
Miike012 said:
The book talks about a unit circle...
360 deg = 2(pi) rad

if it wasnt a unit circle... say r = 4
would it then be
360 deg = 8(pi) rad?

Angles follow what is known as "clock" arithmetic.

It works like a clock would. A clock has 60 minutes and when you go from 59 to 60 you actually go back to "0" on the clock.

Angles work the same way. You have a revolution of 2(pi) radians. When you complete a full revolution you are back to 0 radians. So let's say your angle is 4(pi). Basically think about moving your "clock hand" from 0 to 2(pi) which takes you to zero and then another 2(pi) which takes you to zero again.

The difference with the clock is that the angle starts at the y-axis and goes clockwise while angles in geometry start from the x-axis and go around "counter-clockwise".
 
From Miike012:
Then if it was 4 = r
it would be 8(pi) rad = 360 deg?
Also.. what is the point of telling us that the radius equals the arc? This isn't exactly how it is stated but I think you know what I am talking about.. if not... ill find the exact wording.

We make use of the unit circle, for which the radius is 1, because the circumference is simply 2 \pi

Think again about rotating the 1 unit segment with one endpoint at the circle's center and the other endpoint on the circle. One rotation going around the circle 1 time with sweep through 2 \pi radians.
 
  • #10
Im not sure if anyone has answered my question...
If it was radius 9 or what ever
it would be
360deg = 18(pi) rad?
 
  • #11
What is the definition of a radian?
 
  • #12
Miike012 said:
Nevermind... it would still be 180 deg = 2(pi) rad no matter the radius... Right?

YES (I don't know why somebody told you no). Do you understand why? To understand why, answer sjb-2812's question.

EDIT: Sorry, I understand why somebody told you no...because it's 360 deg that equals 2*pi radians. But the point is that the angle corresponding to a full rotation is 2*pi radians regardless of the radius of the circle over which the rotation takes place.
 
  • #13
I think I got it... one sec
 
  • #14
The angle is the same no matter what the radius of the circle: 2\pi radians is 360 degrees for any circle.

The "radian" measure can be defined as "the arclength the angle cuts of the unit circle". For a general circle, that is "the arclength divided by the radius of the circle".

For 1/4 of a circle, the central angle is 90 degrees or 2\pi/4= \pi/2 radians. "90 degrees" and "\pi/2 radians" measure the same angle. That has nothing to do with radius of the circle or, indeed, whether there is any circle at all.

Also.. what is the point of telling us that the radius equals the arc? This isn't exactly how it is stated but I think you know what I am talking about.. if not... ill find the exact wording.
No, that is not at all true. The "arc equals the radius" only for an angle of 1 radian.
You may be thinking of what I said before: for a circle with radius 1, the arclength equals the measure of the angle in radians.
 
  • #15
My attachment shows two circles... one smaller inside a bigger... with same center...

This shows that the rotation of 2(pi) is the same no matter the radius... is this correct?
 

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  • #16
Miike012 said:
My attachment shows two circles... one smaller inside a bigger... with same center...

This shows that the rotation of 2(pi) is the same no matter the radius... is this correct?

That shows it diagrammatically, but I'll be truly convinced you understand when you express the idea in words or algebraically, by answering sjb-2812's question in post #11. :-p
 
  • #17
Ok I will answer that In a second... I just need this answered...
Please look at this...

Watch at 1 min 29 sec...
She says the two radiuses equal the arc...


The reason why they mention this ( in my opinion) is they want to show us a portion that rotates around the circle that will result in a whole number (2(pi))...

If my opinion is wrong... just say no... and then explain what it meas please...
 
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  • #18
What I think a radian is...
(considering that I know that 360 deg = 2(pi) rad)
radian is symbolic to deg...
radian is another way to explain deg or the rotation around the circle...
Correct or incorrect?
 
  • #19
Miike012 said:
radian is another way to explain deg or the rotation around the circle...

A radian is a different unit of measurement than the degree. Just like meters and feet.
 
  • #20
gb7nash said:
A radian is a different unit of measurement than the degree. Just like meters and feet.

yes... but they are symbolic to one another... they still measure

and just like deg measures around a circle... so do rad
 
  • #21
Radian is the plural form of radius.

Technically, you're measuring the circumference of the circle and your unit of measure is radians (or radiuses if that form for making a noun plural is more easily understandable).

Regardless of the size of the radius, it will take 2pi radians (or radiuses) to complete one circle.

Based on that concept, if lay out a length of one radius on the circumference of a circle, it will cover the same angle regardless of how big or how small the radius happens to be.

If the radius is 1 meter, then 1 meter around the circumference will sweep out 57.3 degrees. If the radius is 5 meters, then 5 meters around the circle will sweep out 57.3 degrees. And so on.

If I measure out 2 radians (radiuses) along the circumference, then it sweeps out 114.6 degrees (2 * 360/(2pi) or 2 * 180/pi since both 360 and 2 are divisible by 2)
 
  • #22
Can you consider this more than sufficiently answered now? I gave you a mechanical/graphical way to understand, and HallsofIvy gave you a good clear practical answer.

You might find more of what you want right here: http://en.wikipedia.org/wiki/Radian
 
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