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

In summary, the discussion centers around the definition of the unit circle and why its radius is chosen to be 1. Some argue that using a circle of radius 1 simplifies the math and makes it easier to teach students about trigonometric functions, while others argue that it introduces unnecessary complications with the factor of 2. There is also mention of the debate between using pi versus tau as the fundamental constant in mathematics. Ultimately, it is suggested to have multiple memberships for the PF member @etotheipi, including @etothetau and @etothephi.
  • #1
sunny79
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8
Why is it that the radius of the unit circle is 1?
 
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  • #2
By definition? What do you think the "unit" in "unit circle" stands for?
 
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  • #3
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
Math_QED said:
By definition? What do you think the "unit" in "unit circle" stands for?
Unit with radius 1
 
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  • #5
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
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
sunny79 said:
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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  • #8
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  • #9
jedishrfu said:
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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  • #10
A.T. said:
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?
 
  • #11
He’s a ##(\tau)## Tau-ist.
 
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  • #12
sysprog said:
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
A.T. said:
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
A.T. said:
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
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
jedishrfu said:
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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  • #17
True, they mention it in the Sciam article.
 
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  • #18
My only argument in favor of ##\pi## is that two pies are better than one.
 
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  • #19
sysprog said:
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
jedishrfu said:
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
sysprog said:
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
But required to post everything twice.
 
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  • #23
etotheipi said:
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
sysprog said:
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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  • #25
Mark44 said:
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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  • #26
I didn't read the links. But it seems to me the question is, is a circle defined by its radius, or by its diameter? There is no definitive answer. If you are drawing a circle (say with a compass) then you think, "radius." If you are measuring a circle (with a ruler or calipers) then you think, "diameter."
 
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  • #27
gmax137 said:
I didn't read the links. But it seems to me the question is, is a circle defined by its radius, or by its diameter? There is no definitive answer. If you are drawing a circle (say with a compass) then you think, "radius." If you are measuring a circle (with a ruler or calipers) then you think, "diameter."
Isn't it true that if we are to measure the circle by use of progressive caliper measurements whereby to determine whether our diametrical measurement is maximal then we have to do that with at least two different pairs of circumferential points in order to by the intersection of the line segments between the thereby determined pairs of points find the center?
 
  • #28
sunny79 said:
Why is it that the radius of the unit circle is 1?

Others have already pointed out that this is just a matter of convenience. I'll just add that early trigonometry (and spherical trigonometry) used a radius of 60 (see Ptolemy's Almagest). That was a convention left over from the Babylonian astronomers. I'm not sure when the unit circle was first popularized. It didn't come up readily in after one minute of searching online. If I had to guess, I'd put my money on Euler having something to do with it.
 
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  • #29
sysprog said:
Isn't it true that if we are to measure the circle by use of progressive caliper measurements whereby to determine whether our diametrical measurement is maximal then we have to do that with at least two different pairs of circumferential points in order to by the intersection of the line segments between the thereby determined pairs of points find the center?
I'm not sure what you're getting at here. I meant calipers like this (with parallel jaws). Squeeze and read the diameter. Much easier than measuring the radius of a given circle.
IMG_20200921_082507964smaller.jpg
 
  • #30
I believe the OP has, quite reasonably, fled in terror...
 
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  • #31
@hutchphd! Still here...just terribly busy.
 
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  • #32
gmax137 said:
I'm not sure what you're getting at here. I meant calipers like this (with parallel jaws). Squeeze and read the diameter. Much easier than measuring the radius of a given circle.
View attachment 269823
Oh, heck, I thought of regular (not necessarily with the spring and screw-wheel apparatus) outside calipers like this:

1601392725965.png
 
  • #33
sysprog said:
Oh, heck, I thought of regular (not necessarily with the spring and screw-wheel apparatus) outside calipers like this:

View attachment 270179
This type of caliper would work like the Vernier caliper shown in post #29. If the jaws of this caliper are set too close, the circle wouldn't fit between the jaws. Opening the jaws to a width so that they just barely accept the circle would give the diameter.
I'm assuming we have an object with circular cross section here, although you could also do the measurement reasonably well for a circle drawn on paper. Put one jaw at any point on the circumference, and adjust the caliper opening so that the other jaw intersects a single point as the caliper is rotated through an arc.
 
  • #34
I'm sorry, my post #26 was a distraction, intending to address the distraction that started around post #9, debating radius vs. diameter, and pi vs. 2*pi.
 
  • #35
Mark44 said:
Opening the jaws to a width so that they just barely accept the circle would give the diameter.
Isn't that still a series of trials? How do we find that we haven't exceeded the diameter? Don't we have to do repeated trials to find out exactly where "just barely" is?
 

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