section_path: Analytic results/Radian
section_name: Radian
paragraph_text: Radian
  
  If a circle of radius r is centred at the vertex of an angle, and that angle intercepts an arc of the circle with an arc length of s, then the radian measure 𝜃 of the angle is the ratio of the arc length to the radius: θ = s r. {\displaystyle \theta ={\frac {s}{r}}.}
  
  The circular arc is said to subtend the angle at the centre of the circle. The angle subtended by a complete circle at its centre is a complete angle, which measures 2 π radians, 360 degrees, or one turn.
  
  Using radians, the formula for the arc length s of a circular arc of radius r and subtending an angle of measure 𝜃 is s = θ r, {\displaystyle s=\theta r,}
  
  and the formula for the area A of a circular sector of radius r and with central angle of measure 𝜃 is A = 1 2 θ r 2. {\displaystyle A={\frac {1}{2}}\theta r^{2}.}
  
  In each formula, the special case 𝜃 = 2 π yields the circumference of a complete circle and area of a complete disk, respectively.