# 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?

Math_QED
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By definition? What do you think the "unit" in "unit circle" stands for?

Abhishek11235, sysprog, berkeman and 1 other person
jedishrfu
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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

Abhishek11235, Hemant, sysprog and 2 others
By definition? What do you think the "unit" in "unit circle" stands for?

Abhishek11235, sysprog, jedishrfu and 1 other person
Infrared
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Another related advantage is that the length of an arc is equal to the angle it subtends (measured in radians).

Abhishek11235, sysprog, symbolipoint and 1 other person
DaveE
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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.

Abhishek11235, Vanadium 50, sysprog and 2 others
symbolipoint
Homework Helper
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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.

Abhishek11235 and sysprog
A.T.
symbolipoint and sysprog
A.T.
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.

symbolipoint
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?

jedishrfu
Mentor
He’s a ##(\tau)## Tau-ist.

tworitdash, xAxis and archaic
A.T.
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.

etotheipi
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2019 Award
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?

DaveE
Gold Member
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...)?

jedishrfu
jedishrfu
Mentor
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:
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

etotheipi and jedishrfu
jedishrfu
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True, they mention it in the Sciam article.

sysprog
jedishrfu
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My only argument in favor of ##\pi## is that two pies are better than one.

tworitdash, rbelli1 and DaveE
etotheipi
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2019 Award
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:

tworitdash, Mark44, sysprog and 1 other person
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?

tworitdash and etotheipi
DaveE
Gold Member
In that case what would you think about letting PF member @etotheipi have a second membership as @etothetau?
Maybe two half memberships?

hutchphd and etotheipi
DaveE
Gold Member
But required to post everything twice.

etotheipi
A.T.
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.

etotheipi
Mark44
Mentor
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.

etotheipi
Based on his post, @etothei1.5pi would be more appropriate.
##-## otherwise rendered as @etothepau ##-## why not let him have 3 memberships?

tworitdash