Coconote
AI notes
AI voice & video notes
Try for free
📐
Understanding Sector Area and Speed
Mar 18, 2025
Lecture Notes: Area of Sector, Linear Speed, and Angular Speed
Area of a Sector of a Circle
Formula
: The area ( A ) of a sector of a circle with radius ( r ) and central angle ( \theta ) in radians is given by: [ A = \frac{1}{2} r^2 \theta ]
Important Note
: ( \theta ) must be in radians. Convert degrees to radians when necessary.
Conversion
: Multiply degrees by ( \frac{\pi}{180} ).
Example Calculation
Problem
: Find the area of a sector with:
Radius: 5 feet
Central angle: 40 degrees
Steps
:
Convert 40 degrees to radians:
( \frac{40 \pi}{180} = \frac{2 \pi}{9} ) radians
Use the formula:
( A = \frac{1}{2} \times 5^2 \times \frac{2 \pi}{9} )
Rounded to two decimal places: 8.73 square feet
Linear Speed
Concept
: Relates to the distance around a circle (arc length).
Formula
: [ v = \frac{s}{t} ]
( v ): Linear speed
( s ): Arc length
( t ): Time
Relation to Distance Formula
: ( s = v \cdot t )
Angular Speed
Concept
: Measured in radians per unit time.
Formula
: [ \omega = \frac{\theta}{t} ]
( \omega ): Angular speed
( \theta ): Central angle in radians
( t ): Time
Example Calculation
Problem
: Object travels around a circle with:
Radius: 5 cm
Central angle: ( \frac{1}{3} ) radians in 20 seconds
Angular Speed
:
( \omega = \frac{1/3}{20} = \frac{1}{60} ) radians/second, or 0.02 radians/second
Linear Speed
:
Calculate arc length ( s ):
( s = \theta \times r = \frac{1}{3} \times 5 = \frac{5}{3} ) cm
Calculate linear speed ( v ):
( v = \frac{5/3}{20} = \frac{1}{12} ) cm/second, or 0.08 cm/second
Important Notes
Ensure units are consistent (e.g., radians for angles).
Check instructions for exact answers or decimal approximations.
📄
Full transcript