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

[sunny79]

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

 

[sysprog]

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:

 

[Mark44]

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.

 

[gmax137]

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.

 

[sysprog]

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?

 

[sysprog]

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.

 

[mathwonk]

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?"