High School Geometry - Sector, Arc Length & Radian Measure


  • Recall that a circle is a collection of all points in a plane equidistant from a fixed point.

Degree - Radian Conversion

  • Degree and radian are the two units that are used to represent the measure of an angle.
  • The relation between degree and the radian is summarized below.
    • 180°=π radians
    • Angle measure in degrees =180π×Angle measure in radians
    • Angle measure in radians =π180×Angle measure in degrees

Length of Arc

  • An arc is a continuous piece of a circle, as shown in the figure below.

  • The length of the arc (minor arc) can be found using the formula mentioned below.
    •  Larc=θ360°×2πr, where θ is in degrees,
    • Larc=r·θ, where θ is in radians

Area of Sector

  • The area of the Sector is the portion (part) of the circular region enclosed by two radii and the corresponding arc, as shown below.

  • The area of a sector (minor sector) can be found using the formula mentioned below.
    •  Asector=θ360°×πr2, where θ is in degrees,
    • Asector=12r2·θ, where θ is in radians

Solved Examples

Example 1: Convert π9 radians into degree measure.

Solution: π radians=180°

1 radian =180°π

π9 radian =π9×180°π=20°

Example 2: Convert 60° to radian measure.

Solution: 180°=π radians

1°=π180 radians

60°=60×π180=π3 radians

Example 3: A circle has a radius of 1 meter. If an arc of this circle subtends an angle 45° at the center O, find the length of the arc.

Solution: Larc=θ360°×2πr


r=1 m

Larc=45°360°×2π×1=π4 m

Example 4: Circle O has radius OP that measures 14 inches with a central angle POQ, where mPOQ=60°. Find the area of the shaded sector.

Solution: Asector=θ360°×πr2

θ=60°, r=14 in

Asector=60°360°×227×142=102.67 in2

Cheat Sheet

  • Degree - Radian relation: 180°=π radians
  • Arc length:
    •  Larc=θ360°×2πr, where θ is in degrees
    • Larc=r·θ, where θ is in radians
  • Area of a sector:
    •  Asector=θ360°×πr2, where θ is in degrees
    • Asector=12r2·θ, where θ is in radians
  • The measure of an inscribed angle is half the measure of the intercepted arc.

  • The angle in a semicircle is a right angle.

Blunder Area

  • While calculating the length of the arc or area of the sector, pay attention to using the appropriate formulas depending on the unit of measure of the angle.
  • When the diameter of a circle is given, sometimes students put diameter in place of radius in the formulas.