- #1

sunny79

- 77

- 8

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

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- B
- Thread starter sunny79
- Start date

- #1

sunny79

- 77

- 8

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

- #2

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

- #3

jedishrfu

Mentor

- 14,061

- 8,019

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

- #4

sunny79

- 77

- 8

Unit with radius 1By definition? What do you think the "unit" in "unit circle" stands for?

- #5

Infrared

Science Advisor

Gold Member

- 965

- 538

- #6

DaveE

Science Advisor

Gold Member

- 2,710

- 2,371

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.

- #7

symbolipoint

Homework Helper

Education Advisor

Gold Member

- 6,904

- 1,573

Excellent responses given so far, but the question is silly.Why is it that the radius of the unit circle is 1?

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.

- #8

A.T.

Science Advisor

- 11,545

- 2,909

Because 1 is the neutral element of multiplication, which simplifies a lot of the math:Why is it that the radius of the unit circle is 1?

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

- #9

A.T.

Science Advisor

- 11,545

- 2,909

While using a circle of radius 1 provides a simplification, introducing a constant that isThe 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.

- #10

sysprog

- 2,613

- 1,783

Can you please elaborate further on why you think that the ##2\pi## radians is an unnecessary complication?While using a circle of radius 1 provides a simplification, introducing a constant that isjust halfof its perimeter, makes it unnecessarily complicated again.

- #11

jedishrfu

Mentor

- 14,061

- 8,019

He’s a ##(\tau)## Tau-ist.

- #12

A.T.

Science Advisor

- 11,545

- 2,909

Beacause of the unnecessary factor 2. It's like using a circle with radius 1/2 instead of the unit circle.Can you please elaborate further on why you think that the ##2\pi## radians is an unnecessary complication?

- #13

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

Science Advisor

Gold Member

- 2,710

- 2,371

Are you saying π should have been defined as circumference/radius (π=6.283...)?While using a circle of radius 1 provides a simplification, introducing a constant that isjust halfof its perimeter, makes it unnecessarily complicated again.

- #15

jedishrfu

Mentor

- 14,061

- 8,019

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.

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

sysprog

- 2,613

- 1,783

I think that this is a nice article on the topic: https://tauday.com/tau-manifestoSciam 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.

- #17

jedishrfu

Mentor

- 14,061

- 8,019

True, they mention it in the Sciam article.

- #18

jedishrfu

Mentor

- 14,061

- 8,019

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

- #19

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:

- #20

sysprog

- 2,613

- 1,783

In that case what would you think about letting PF member @etotheipi have a second membership as @etothetau?My only argument in favor of ##\pi## is that two pies are better than one.

- #21

DaveE

Science Advisor

Gold Member

- 2,710

- 2,371

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

- #22

DaveE

Science Advisor

Gold Member

- 2,710

- 2,371

But required to post everything twice.

- #23

A.T.

Science Advisor

- 11,545

- 2,909

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.What if you want to write the area of the circle?

- #24

Mark44

Mentor

- 36,468

- 8,437

Based on his post, @etothei1.5pi would be more appropriate.In that case what would you think about letting PF member @etotheipi have a second membership as @etothetau?

- #25

sysprog

- 2,613

- 1,783

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

- #26

gmax137

Science Advisor

- 2,209

- 1,825

- #27

sysprog

- 2,613

- 1,783

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

brainpushups

- 437

- 186

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

- #29

gmax137

Science Advisor

- 2,209

- 1,825

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.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?

- #30

hutchphd

Science Advisor

Homework Helper

- 5,152

- 4,329

I believe the OP has, quite reasonably, fled in terror...

- #31

sunny79

- 77

- 8

@hutchphd! Still here...just terribly busy.

- #32

sysprog

- 2,613

- 1,783

Oh, heck, I thought of regular (not necessarily with the spring and screw-wheel apparatus) outside calipers like this: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

- #33

Mark44

Mentor

- 36,468

- 8,437

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.Oh, heck, I thought of regular (not necessarily with the spring and screw-wheel apparatus) outside calipers like this:

View attachment 270179

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

gmax137

Science Advisor

- 2,209

- 1,825

- #35

sysprog

- 2,613

- 1,783

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?Mark44 said:Opening the jaws to a width so that they just barely accept the circle would give the diameter.

Share:

- Last Post

- Replies
- 5

- Views
- 1K

- Last Post

- Replies
- 2

- Views
- 424

- Replies
- 8

- Views
- 389

- Last Post

- Replies
- 3

- Views
- 232

- Last Post

- Replies
- 10

- Views
- 514

- Last Post

- Replies
- 3

- Views
- 458

- Last Post

- Replies
- 16

- Views
- 535

- Last Post

- Replies
- 1

- Views
- 370

- Last Post

- Replies
- 2

- Views
- 619

- Replies
- 6

- Views
- 275