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Correct. That's why radians are defined as a length on the unit circle. Degrees are the same. Given the radius, a degree or a radian always denotes a particular length around the circumference.
For each degree of a circle there are two lengths, the length of the arc and the length of the cord. The ratio of any arc to a cord is 1.1107207345.
An angle has a distance between two different points in which the angle ends. All angles, hence degrees, result in lengths. The degree, as a length, is 1/360th of a circle's circumferance. With the circumferance unraveled into a straight line the degree thus results in 1/360 of a line.RCSaunders wrote: ↑Mon Jun 07, 2021 9:36 pmFor each degree of a circle there are two lengths, the length of the arc and the length of the cord. The ratio of any arc to a cord is 1.1107207345.
For every the length of one degree arc is 1/360 X 2πr, which means the length of any arc for the same angle is different for every circle with a different radius. There is no direct relationship between length and angle.
This will help: Length is a measure distance. Angle is a measure of direction.
They are totally different things. Now go back and study your geometry.
It's not a paradox. Radians are defined with respect to the unit circle; and if you want to define a degree as a length, you should likewise normalize it to the unit circle. But note that the ratio of the length along the circumference to the radius is always the same for any angle, so if you define a radian or a degree in terms of circumference/radius you'll always get the same number.Eodnhoj7 wrote: ↑Mon Jun 07, 2021 10:40 pm
An angle has a distance between two different points in which the angle ends. All angles, hence degrees, result in lengths. The degree, as a length, is 1/360th of a circle's circumferance. With the circumferance unraveled into a straight line the degree thus results in 1/360 of a line.
The paradox is if one line is longer than another then 1/360th of the line is longer than another 1/360th of the line. The degree thus results in unequal lengths yet all degrees result in lengths. The degree has no constant length yet is defined as 1/360th of a line regardless.
Direction always results in a distance as the direction is the pointing of one point to another point. Direction and distance are interwoven.
To be fair, a radian is a directed length.RCSaunders wrote: ↑Mon Jun 07, 2021 9:36 pm This will help: Length is a measure distance. Angle is a measure of direction.
The ratio may always be the same but the lengths vary. There is no constant length which results from the degree yet the degree is always the same. If the degree is to have a constant length it must have a constant ratio, of course, but the lengths must always be equal.wtf wrote: ↑Mon Jun 07, 2021 10:57 pmIt's not a paradox. Radians are defined with respect to the unit circle; and if you want to define a degree as a length, you should likewise normalize it to the unit circle. But note that the ratio of the length along the circumference to the radius is always the same for any angle, so if you define a radian or a degree in terms of circumference/radius you'll always get the same number.Eodnhoj7 wrote: ↑Mon Jun 07, 2021 10:40 pm
An angle has a distance between two different points in which the angle ends. All angles, hence degrees, result in lengths. The degree, as a length, is 1/360th of a circle's circumferance. With the circumferance unraveled into a straight line the degree thus results in 1/360 of a line.
The paradox is if one line is longer than another then 1/360th of the line is longer than another 1/360th of the line. The degree thus results in unequal lengths yet all degrees result in lengths. The degree has no constant length yet is defined as 1/360th of a line regardless.
Direction always results in a distance as the direction is the pointing of one point to another point. Direction and distance are interwoven.
To be fair, a radian is a directed length.RCSaunders wrote: ↑Mon Jun 07, 2021 9:36 pm This will help: Length is a measure distance. Angle is a measure of direction.
The real mistake is in the assumption an angle only pertains to degrees in a circle (a method of defining angles in a plane, but the length of a radian (or a radius) is irrelevant to an angle. Lenth is only the distance between two points. There is no logical requirement for any angle for there to be length. It's a confusion of concepts.wtf wrote: ↑Mon Jun 07, 2021 10:57 pmIt's not a paradox. Radians are defined with respect to the unit circle; and if you want to define a degree as a length, you should likewise normalize it to the unit circle. But note that the ratio of the length along the circumference to the radius is always the same for any angle, so if you define a radian or a degree in terms of circumference/radius you'll always get the same number.Eodnhoj7 wrote: ↑Mon Jun 07, 2021 10:40 pm
An angle has a distance between two different points in which the angle ends. All angles, hence degrees, result in lengths. The degree, as a length, is 1/360th of a circle's circumferance. With the circumferance unraveled into a straight line the degree thus results in 1/360 of a line.
The paradox is if one line is longer than another then 1/360th of the line is longer than another 1/360th of the line. The degree thus results in unequal lengths yet all degrees result in lengths. The degree has no constant length yet is defined as 1/360th of a line regardless.
Direction always results in a distance as the direction is the pointing of one point to another point. Direction and distance are interwoven.
To be fair, a radian is a directed length.RCSaunders wrote: ↑Mon Jun 07, 2021 9:36 pm This will help: Length is a measure distance. Angle is a measure of direction.
360 degrees result in line divided into 360 segments when the circumferance is unraveled (be it a circle or 360-gon). All angles have a distance between their two end points thus a length is necessary.RCSaunders wrote: ↑Tue Jun 08, 2021 1:42 amThe real mistake is in the assumption an angle only pertains to degrees in a circle (a method of defining angles in a plane, but the length of a radian (or a radius) is irrelevant to an angle. Lenth is only the distance between two points. There is no logical requirement for any angle for there to be length. It's a confusion of concepts.wtf wrote: ↑Mon Jun 07, 2021 10:57 pmIt's not a paradox. Radians are defined with respect to the unit circle; and if you want to define a degree as a length, you should likewise normalize it to the unit circle. But note that the ratio of the length along the circumference to the radius is always the same for any angle, so if you define a radian or a degree in terms of circumference/radius you'll always get the same number.Eodnhoj7 wrote: ↑Mon Jun 07, 2021 10:40 pm
An angle has a distance between two different points in which the angle ends. All angles, hence degrees, result in lengths. The degree, as a length, is 1/360th of a circle's circumferance. With the circumferance unraveled into a straight line the degree thus results in 1/360 of a line.
The paradox is if one line is longer than another then 1/360th of the line is longer than another 1/360th of the line. The degree thus results in unequal lengths yet all degrees result in lengths. The degree has no constant length yet is defined as 1/360th of a line regardless.
Direction always results in a distance as the direction is the pointing of one point to another point. Direction and distance are interwoven.
To be fair, a radian is a directed length.RCSaunders wrote: ↑Mon Jun 07, 2021 9:36 pm This will help: Length is a measure distance. Angle is a measure of direction.
Yes it does the endpoints are the opposite ends to where the two lines unite.
From HomeSchoolMathEodnhoj7 wrote: ↑Tue Jun 08, 2021 4:15 pmYes it does the endpoints are the opposite ends to where the two lines unite.
So does a line contain infinite lines yet heads off to infinity...but it still has an end.RCSaunders wrote: ↑Tue Jun 08, 2021 9:49 pmFrom HomeSchoolMath
"A ray starts out at a point and continues off to infinity."
An angle is made up of two rays that have the same beginning point.
There are no end points to an angle.
Your thinking of a triangle, if you're thinking at all.
Projection...considering you have not given an argument explaining your view....and to get back on track: