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

  • B
  • Thread starter sunny79
  • Start date
  • #1
77
8
Why is it that the radius of the unit circle is 1?
 

Answers and Replies

  • #2
Math_QED
Science Advisor
Homework Helper
2019 Award
1,701
720
By definition? What do you think the "unit" in "unit circle" stands for?
 
  • Like
  • Haha
Likes Abhishek11235, sysprog, berkeman and 1 other person
  • #3
12,082
5,752
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
 
  • Like
Likes Abhishek11235, Hemant, sysprog and 2 others
  • #4
77
8
By definition? What do you think the "unit" in "unit circle" stands for?
Unit with radius 1
 
  • Like
Likes Abhishek11235, sysprog, jedishrfu and 1 other person
  • #5
Infrared
Science Advisor
Gold Member
788
408
Another related advantage is that the length of an arc is equal to the angle it subtends (measured in radians).
 
  • Like
Likes Abhishek11235, sysprog, symbolipoint and 1 other person
  • #6
DaveE
Gold Member
878
638
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.
 
  • Like
  • Informative
Likes Abhishek11235, Vanadium 50, sysprog and 2 others
  • #7
symbolipoint
Homework Helper
Education Advisor
Gold Member
6,054
1,128
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.
 
  • Like
Likes Abhishek11235 and sysprog
  • #9
A.T.
Science Advisor
10,640
2,231
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.
 
  • Like
Likes symbolipoint
  • #10
1,649
905
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
12,082
5,752
He’s a ##(\tau)## Tau-ist.
 
  • Like
  • Haha
Likes vela, tworitdash, xAxis and 1 other person
  • #12
A.T.
Science Advisor
10,640
2,231
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
Gold Member
2019 Award
2,703
1,615
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
Gold Member
878
638
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...)?
 
  • Like
Likes jedishrfu
  • #15
12,082
5,752
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.
 
Last edited:
  • #16
1,649
905
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
 
  • Like
Likes etotheipi and jedishrfu
  • #17
12,082
5,752
True, they mention it in the Sciam article.
 
  • Like
Likes sysprog
  • #18
12,082
5,752
My only argument in favor of ##\pi## is that two pies are better than one.
 
  • Haha
  • Like
Likes tworitdash, rbelli1 and DaveE
  • #19
etotheipi
Gold Member
2019 Award
2,703
1,615
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
 
  • Haha
  • Like
Likes tworitdash, Mark44, sysprog and 1 other person
  • #20
1,649
905
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? 🤔
 
  • Haha
  • Love
Likes tworitdash and etotheipi
  • #21
DaveE
Gold Member
878
638
In that case what would you think about letting PF member @etotheipi have a second membership as @etothetau? 🤔
Maybe two half memberships?
 
  • Like
Likes hutchphd and etotheipi
  • #22
DaveE
Gold Member
878
638
But required to post everything twice.
 
  • Wow
Likes etotheipi
  • #23
A.T.
Science Advisor
10,640
2,231
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.
 
  • Like
Likes etotheipi
  • #24
34,149
5,764
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:
 
  • Like
Likes etotheipi
  • #25
1,649
905
Based on his post, @etothei1.5pi would be more appropriate. :oldbiggrin:
##-## otherwise rendered as @etothepau ##-## why not let him have 3 memberships? :cool:
 
  • Haha
Likes tworitdash
Top