I will remeber this
thanks

Don't "remember" it. UNDERSTAND IT!

Part of your problem is that you have a hard time putting in the effort to learn concepts and, instead, just seem to want to memorize specific facts that are often application to a specific problem. This is crippling your ability to move forward. You need to learn the concepts and then how to apply those concepts to new problems.

 

Yep. The Radian is also a unitless number.

Why?

π

π=22/7 as i was taught it is constant
or The number π is a mathematical constant, the ratio of a circle's circumference to its diameter, commonly approximated as 3.14159.

The reason it is dimensionless is explained directly by the what you say you were taught about it -- that it is it a ratio of two lengths, hence the units cancel leaving you with a dimensionless number.

Do you see what I mean about learning concepts instead of memorizing facts? You were taught that, "π is a mathematical constant, the ratio of a circle's circumference to its diameter," but you treat this is just one disembodied fact to be memorized and regurgitated. Thus when pondering why radians have no units you are unable to apply the concept embodied by what the radian measure is, namely the ratio of the length of the arc subtended by an angle to the radius of that arc, to figure out the answer to your question.

 

The reason it is dimensionless is explained directly by the what you say you were taught about it -- that it is it a ratio of two lengths, hence the units cancel leaving you with a dimensionless number.

Do you see what I mean about learning concepts instead of memorizing facts? You were taught that, "π is a mathematical constant, the ratio of a circle's circumference to its diameter," but you treat this is just one disembodied fact to be memorized and regurgitated. Thus when pondering why radians have no units you are unable to apply the concept embodied by what the radian measure is, namely the ratio of the length of the arc subtended by an angle to the radius of that arc, to figure out the answer to your question.

 

There is a nice picture at www.mathopenref.com/radians.html

Touch the circle and see the angle stay the same irrespective of circle size. You will see the arc (length r) is always the same as the radius (length r).

Assume they are in cm. Then that angle, in radians, is r cm/r cm which is 1. The r's cancel and the cm cancel so only the ratio is left. So the unit of that angle you see on that web page is not: feet, light years, degrees, farenheit, decibels, fathoms, pecks, miles or gallons. It is radians, and in that case on the web page, there is just one of them highlighted.