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

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The radius of a unit circle is defined as 1 for simplicity in teaching trigonometry and radians, allowing for easier calculations, such as the circle's perimeter being 2π radians. This definition also facilitates the use of sine and cosine functions, as right triangles inscribed within the circle have hypotenuses equal to the radius. While some argue that using a radius of 1 simplifies mathematical concepts, others suggest that it can complicate understanding by introducing unnecessary constants. The discussion touches on historical conventions in trigonometry, noting that earlier systems used different radius values, such as 60. Ultimately, the choice of a unit circle with a radius of 1 is primarily a matter of convenience in mathematical practice.
  • #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?
 
  • #36
Mark44 said:
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.
This method makes good sense to me.
 
  • #37
of course the assumption that caliper measure equals ruler measure is equivalent to the Side angle side axiom of euclidean geometry.

Pardon me, I am entranced by a thread that goes on this long, in answer to OP's question of "why is 1 = 1?"
 
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