Why does the radius of a unit circle need to be 1?

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Why is it that the radius of the unit circle is 1?
 

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  • #2
Math_QED
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By definition? What do you think the "unit" in "unit circle" stands for?
 
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  • #3
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Of course, the follow on question would be why do mathematicians define it that way?

The benefits are that it provides a simplification when teaching students about the trig functions and radians. Using a 1 means the circle perimeter is now ##2\pi## radians.

Right triangles drawn inside the circle with their hypotenuses being the radius have sides that sin and cos measurements their acute angles.

Im sure there’s other benefits as well. Can you spot any?

https://en.wikipedia.org/wiki/Unit_circle
 
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  • #4
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By definition? What do you think the "unit" in "unit circle" stands for?
Unit with radius 1
 
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  • #5
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Another related advantage is that the length of an arc is equal to the angle it subtends (measured in radians).
 
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  • #6
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I think you are misunderstanding what people are saying when they talk about unit circles. This is a definition, there is no inherent reason for it. It's just a different way of saying r=1 (because it's so common, it has a name).

I think this is the same as asking why does a circle with radius =13 have radius =13? They've just used different words for r=1.
 
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  • #7
symbolipoint
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Why is it that the radius of the unit circle is 1?
Excellent responses given so far, but the question is silly.

Further Thought: My hasty thinking to say, "silly". One can look at a few measurable parts of a circle. circumference, diameter, radius, area. To pick RADIUS of unit 1 allows for some ease in handling some Trigonometry.
 
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  • #9
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The benefits are that it provides a simplification when teaching students about the trig functions and radians. Using a 1 means the circle perimeter is now ##2\pi## radians.
While using a circle of radius 1 provides a simplification, introducing a constant that is just half of its perimeter, makes it unnecessarily complicated again.
 
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While using a circle of radius 1 provides a simplification, introducing a constant that is just half of its perimeter, makes it unnecessarily complicated again.
Can you please elaborate further on why you think that the ##2\pi## radians is an unnecessary complication?
 
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He’s a ##(\tau)## Tau-ist.
 
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  • #12
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Can you please elaborate further on why you think that the ##2\pi## radians is an unnecessary complication?
Beacause of the unnecessary factor 2. It's like using a circle with radius 1/2 instead of the unit circle.
 
  • #13
etotheipi
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Beacause of the unnecessary factor 2. It's like using a circle with radius 1/2 instead of the unit circle.
What if you want to write the area of the circle?
 
  • #14
DaveE
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While using a circle of radius 1 provides a simplification, introducing a constant that is just half of its perimeter, makes it unnecessarily complicated again.
Are you saying π should have been defined as circumference/radius (π=6.283...)?
 
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  • #15
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Sciam did a nice article on the pros and cons:

https://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/#:~:text=At its heart, pi refers,now a proponent of tau.

I am fearful though that it will become another political football as the definition of pi=3 almost did many years ago in Indiana:

https://en.wikipedia.org/wiki/Indiana_Pi_Bill

Something like this happened with the change of notation between physics and math over the spherical coordinate system.

I was taught in the early 1970s ##R \theta \phi## (##\phi## for the angle with the z-axis) whereas the physics usage at work was ##R \phi \theta## (##\theta## with the z-axis) . At first, I thought my brain was losing it until I did some research and discovered I was taught the math convention.

You can imagine the confusion that results in trying to understand any spherically symmetric physical systems.
 
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  • #16
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Sciam did a nice article on the pros and cons:

https://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/#:~:text=At its heart, pi refers,now a proponent of tau.

I am fearful though that it will become another political football as the definition of pi=3 almost did many years ago in Indiana:

https://en.wikipedia.org/wiki/Indiana_Pi_Bill

Something like this happened with the change of notation between physics and math over the spherical coordinate system. I was taught in the early 1970s ##R \theta \phi## (##\phi## for the angle with the z-axis)whereas physics usage at work was ##R \phi \theta## (##\theta## with the z-axis) . You can imagine the confusion that results in trying to understand any spherically symmetric physical systems.
I think that this is a nice article on the topic: https://tauday.com/tau-manifesto
 
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True, they mention it in the Sciam article.
 
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  • #18
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My only argument in favor of ##\pi## is that two pies are better than one.
 
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  • #19
etotheipi
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I think that this is a nice article on the topic: https://tauday.com/tau-manifesto
That's a really fun page; I noticed that they justified ##A = \frac{1}{2} \tau r^2## by analogy for other quadratic forms that arise in Physics. The stuff about Gaussian distributions and polar coordinates is a nice touch. Perhaps we can agree on:

1600280268017.png
 
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  • #20
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My only argument in favor of ##\pi## is that two pies are better than one.
In that case what would you think about letting PF member @etotheipi have a second membership as @etothetau? 🤔
 
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  • #21
DaveE
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In that case what would you think about letting PF member @etotheipi have a second membership as @etothetau? 🤔
Maybe two half memberships?
 
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  • #22
DaveE
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But required to post everything twice.
 
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  • #23
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What if you want to write the area of the circle?
How do you write the area of a triangle? The circle area can be derived from that, so it makes sense for them to have a similar form.
 
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  • #24
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In that case what would you think about letting PF member @etotheipi have a second membership as @etothetau? 🤔
Based on his post, @etothei1.5pi would be more appropriate. :oldbiggrin:
 
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Based on his post, @etothei1.5pi would be more appropriate. :oldbiggrin:
##-## otherwise rendered as @etothepau ##-## why not let him have 3 memberships? :cool:
 
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