Why are circles infinitely smooth if they have degrees?

[Cody Richeson]

Because a triangle comes out to 180 degrees, and yet it can only have three sides. A circle has 360 degrees, but its number of "sides" are uncountable. Can someone explain this?

 

[phinds]

Because a triangle comes out to 180 degrees, and yet it can only have three sides. A circle has 360 degrees, but its number of "sides" are uncountable. Can someone explain this?

In the sense in which a triangle has 360 degrees a circle has an infinite number of degrees so you are comparing apples and oranges.

To see this, just take n-sided figures where n continues to increase and add up all the internal angles just like you do for a triangle. You'll see that as n goes up, so does the sum of the angles so as n approaches infinity the number of degrees approaches infinity.

 

[HallsofIvy]

Because a triangle comes out to 180 degrees, and yet it can only have three sides. A circle has 360 degrees, but its number of "sides" are uncountable. Can someone explain this?

You are comparing apples and oranges. The "180 degrees" in a triangle is the sum of the three vertex angles. The "360 degrees" in a circle are measured at the center of the circle.

 

[Cody Richeson]

You are comparing apples and oranges. The "180 degrees" in a triangle is the sum of the three vertex angles. The "360 degrees" in a circle are measured at the center of the circle.

So any measurement of degrees radiating from a center point will necessarily be round? Makes sense. So this is not the same for triangles, I suppose, or any polygon? What about a polygon with a very large number of sides that almost appears circular?

 

[Mark44]

So any measurement of degrees radiating from a center point will necessarily be round?
Makes sense.

No, not really. If you're measuring degrees, you're measuring an angle that a ray has rotated relative to some fixed direction. The degree measure has nothing to do with the length of the ray.
As an example, suppose you are a surveyor and you look through your transit at the top of a hill in the distance. You notice another hill that is to the left and farther away, and rotate the transit through an angle of 15°. The distances along the two sight lines aren't equal, so you aren't at the center of a circle with the hills on the circumference.

So this is not the same for triangles, I suppose, or any polygon? What about a polygon with a very large number of sides that almost appears circular?

 

[jbriggs444]

So any measurement of degrees radiating from a center point will necessarily be round? Makes sense. So this is not the same for triangles, I suppose, or any polygon? What about a polygon with a very large number of sides that almost appears circular?

If by "round" you mean "adds up to 360" then the answer is yes. They're all 360 degrees.

Position yourself at any point inside the figure. Set up your protractor and measure the angle subtended by the first side. Repeat for the second side. And the third. And so on. If you are positioned exactly in the center of an equilateral triangle, these angles will measure 120, 120 and 120 degrees. Total 360. If you are positioned exactly in the center of a square they will measure 90, 90, 90 and 90. Total 360. This is the same for every other regular polygon measured from its center.

It is also true when measured from any other interior point, not just from the center. If you are measuring from near a corner of a square, for example, you could get close to 45, 45, 135, 135 and the total would still be 360. It is also true for irregular and non-convex polygons. Mark44's example of a surveyor's transit is apt. If you point the transit in turn at each vertex of the polygon and sweep past all of them and back to the first one, you will have covered a net 360 degrees of arc.

[Or 720 or or some other multiple of 360 if the "polygon" goes around more than once -- i.e. if it has a winding number greater than 1]

 

[aikismos]

Every convex polygon inscribed in a circle also has 360 degrees if you add up the measures of the apothems. https://www.mathsisfun.com/definitions/apothem.html