High School Geometry - Sector, Arc Length & Radian Measure

Introduction

• 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.

• • $180°=\mathrm{\pi } \text{radians}$
  • Angle measure in degrees $=\frac{180}{\mathrm{\pi }}\times$Angle measure in radians
  • Angle measure in radians $=\frac{\mathrm{\pi }}{180}\times$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.

• •  $L_{arc}=\frac{\theta }{360°}\times 2\mathrm{\pi r}$, where $\theta$ is in degrees,

• • $L_{arc}=r·\theta$, where $\theta$ 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.
  
  •  $A_{sector}=\frac{\theta }{360°}\times {\mathrm{\pi r}}^2$, where $\theta$ is in degrees,
  • $A_{sector}=\frac{1}{2}{r}^2·\theta$, where $\theta$ is in radians

Solved Examples

Example 1: Convert $\frac{\mathrm{\pi }}{9}$ radians into degree measure.

Solution: $\mathrm{\pi } \text{radians}=180°$

$1 \text{radian} =\frac{180°}{\mathrm{\pi }}$

$\frac{\mathrm{\pi }}{9}\text{ radian} =\frac{\mathrm{\pi }}{9}\times \frac{180°}{\mathrm{\pi }}$$=20°$

Example 2: Convert $60°$ to radian measure.

Solution: $180°=\mathrm{\pi } \text{radians}$

$1°=\frac{\mathrm{\pi }}{180} \text{radians}$

$60°=60\times \frac{\mathrm{\pi }}{180}$$=\frac{\mathrm{\pi }}{3} \text{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: $L_{arc}=\frac{\theta }{360°}\times 2\mathrm{\pi r}$

$\theta =45°$

$r=1 m$

$L_{arc}=\frac{45°}{360°}\times 2\mathrm{\pi }\times 1$$=\frac{\mathrm{\pi }}{4} m$

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

Solution: $A_{sector}=\frac{\theta }{360°}\times {\mathrm{\pi r}}^2$

$\theta =60°, r=14 in$

$A_{sector}=\frac{60°}{360°}\times \frac{22}{7}\times {\left(14\right)}^2=102.67 i{n}^2$

Cheat Sheet

• Degree - Radian relation: $180°=\mathrm{\pi } \text{radians}$
• Arc length:
  •  $L_{arc}=\frac{\theta }{360°}\times 2\mathrm{\pi r}$, where $\theta$ is in degrees
  • $L_{arc}=r·\theta$, where $\theta$ is in radians
• Area of a sector:
  •  $A_{sector}=\frac{\theta }{360°}\times {\mathrm{\pi r}}^2$, where $\theta$ is in degrees
  • $A_{sector}=\frac{1}{2}{r}^2·\theta$, where $\theta$ 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.