By definition,

L (length of arc) = ( \dfrac { \mathrm{θ_{deg} } } {360} ) × Circumference (arc length is proportional to angle, one complete arc subtends 360° at center)

Also, Circumference = 2 𝛑 r

Hence, L = ( \dfrac { \mathrm{θ_{deg} } } {360} ) × 2 𝛑 r      – – – – – (i)

Now, again by definition,  θrad   = ( \dfrac{ \mathrm{L} }{ r } )    (radian is ration of arc length to radius)

So, L = r × θrad    – – – – – (ii)

From (i) & (ii), we have 

r × θrad= ( \dfrac { \mathrm{θ_{deg} } } {360} ) × 2 𝛑 r

Or,

θrad = θdeg  × ( \dfrac {2𝛑}  {360} )