Is the radian always equal to the radius of a circle?

Question

Is the radian of a circle always equal to the radius of a circle? If so, are there always the same number of radians in every circle? If an angle has a measure of 30 degrees, the radian would be around .52. Does that mean that the radius is .52?

Answer 1

The radian measure of an angle, θ, with vertex at the center of a circle of radius r and which cuts off an arc of length s on the circle is defined as θ = s/r.

 

 For example,  a 30° angle cuts off an arc which is one 12th of the circle's circumference, regardless of the circle's radius.

 

So, if r = 1, the radian measure is (1/12)(2π)(1)/1 = π/6

 

     If r = 5, the radian measure is (1/12)(2π)(5)/5 = π/6

 

To answer your question, the radian measure of an angle is not in general the same as the radius of the circle.  However, regardless of the radius, the radian measure of a given angle is always the same.   

 

 

Answer 2

The DEFINITION of radians comes from the alternative measurement system of angle measure, of a central angle of a circle: θ=s/r where s is the length of the subtending (intercepted) angle, and r is the circle's radius.  Consequently, we have the equation 360^o = 2pi radians, or 180^o=pi radians. Therefore 30^o = pi/6 radians. When r=1 ("unit circle") θ=s, the angle's arclength.

 

The answer to Is the radian of a circle always equal to the radius of a circle? is NO--it's actually not a proper question.

Answer 3

Yes, by definition.

Yes, 2π

30° is 1/12 of 2π radians