Notes on Arc Length and Area of a Sector

Arc Length and Area of a Sector

Definitions

  • Arc Length (s): For a circle of radius, rr, and a central angle hetaheta (in radians), the arc length is given by the formula:   s=rhetas = r heta
      where:
      - ss = arc length
      - rr = radius of the circle
      - hetaheta = central angle in radians

Example Calculations for Arc Length

  1. Example 1: Find the arc length for a circle with a radius of 4 inches and a central angle of 240°.    - Convert degrees to radians:
       heta=240extoimesracextπrad180exto=rac4extπ3extradheta = 240^ ext{o} imes rac{ ext{π rad}}{180^ ext{o}} = rac{4 ext{π}}{3} ext{ rad}
       - Compute arc length:
       s=rheta=4imesrac4extπ3=rac16extπ3extinchess = r heta = 4 imes rac{4 ext{π}}{3} = rac{16 ext{π}}{3} ext{ inches}

  2. Example 2: Given r=27r = 27 meters and heta=2heta = 2 radians, find ss.    - s=rheta=27imes2=54extmeterss = r heta = 27 imes 2 = 54 ext{ meters}

  3. Example 3: Given s=36s = 36 feet and heta=hetaheta = heta (in radians), find rr.    - Rearranging s=rhetas = r heta gives r=racsheta=rac36hetaextfeetr = rac{s}{ heta} = rac{36}{ heta} ext{ feet}

  4. Example 4: Given s=82s = 82 miles and heta=135extoheta = 135^ ext{o}, find rr.    - Convert degrees to radians:
       heta=135extoimesracextπrad180exto=rac3extπ4extradheta = 135^ ext{o} imes rac{ ext{π rad}}{180^ ext{o}} = rac{3 ext{π}}{4} ext{ rad}
       - r=racsheta=rac82rac3extπ4=rac82imes43extπextmilesr = rac{s}{ heta} = rac{82}{ rac{3 ext{π}}{4}} = rac{82 imes 4}{3 ext{π}} ext{ miles}

  5. Example 5: For a 15 inch pizza with a 40° central angle, find the crust's length.    - Convert degrees to radians:    heta=40extoimesracextπrad180exto=rac2extπ9extradheta = 40^ ext{o} imes rac{ ext{π rad}}{180^ ext{o}} = rac{2 ext{π}}{9} ext{ rad}    - Compute arc length:    s=rheta=15imesrac2extπ9=rac30extπ9extinchess = r heta = 15 imes rac{2 ext{π}}{9} = rac{30 ext{π}}{9} ext{ inches}

  6. Example 6: For a piece of pumpkin pie with a perimeter of 20 inches and diameter of 12 inches, find the central angle (in degrees and radians).    - The circumference of the pie is:    C=extdiameterimesextπ=12extinimesextπ=12πextinchesC = ext{diameter} imes ext{π} = 12 ext{ in} imes ext{π} = 12π ext{ inches}
       - The length of the arc, if it is a piece of the pie, is:    extarclength=C−(ext2radii)=20−12=8extinchesext{arc length} = C - ( ext{2 radii}) = 20 - 12 = 8 ext{ inches}
       - Use the arc length to find the angle:
       heta=racsr=rac86=rac43extradians extDegrees:rac43imesrac180extoextπ=76extoheta = rac{s}{r} = rac{8}{6} = rac{4}{3} ext{ radians} \ ext{Degrees: } rac{4}{3} imes rac{180^ ext{o}}{ ext{π}} = 76^ ext{o}

Area of a Sector

  • Area (A): The area of a sector of a circle formed by a central angle hetaheta (in radians) is given by:   A=rac12r2hetaA = rac{1}{2}r^2 heta   where:
      - AA = area of the sector
      - rr = radius
      - hetaheta = central angle in radians

Example Calculations for Area of a Sector

  1. Example 1: Given r=4r = 4 feet and heta=1.5heta = 1.5 radians, find AA.    - A=rac12imes(42)imes1.5=rac12imes16imes1.5=12extsquarefeetA = rac{1}{2} imes (4^2) imes 1.5 = rac{1}{2} imes 16 imes 1.5 = 12 ext{ square feet}

  2. Example 2: Given r=10r = 10 miles and heta=rac12heta = rac{1}{2} radians, find AA.    - A=rac12imes(102)imesrac12=rac12imes100imesrac12=25extsquaremilesA = rac{1}{2} imes (10^2) imes rac{1}{2} = rac{1}{2} imes 100 imes rac{1}{2} = 25 ext{ square miles}

  3. Example 3: Given r=7r = 7 km and heta=60extoheta = 60^ ext{o}, find AA.    - Convert degrees to radians:
       heta=60extoimesracextπrad180exto=racextπ3heta = 60^ ext{o} imes rac{ ext{π rad}}{180^ ext{o}} = rac{ ext{π}}{3}    - Compute area:
       A=rac12imes(72)imesracextπ3=rac49extπ6extsquarekmA = rac{1}{2} imes (7^2) imes rac{ ext{π}}{3} = rac{49 ext{π}}{6} ext{ square km}

  4. Example 4: Find the Area for:    - Given r=14r = 14 cm and heta=315extoheta = 315^ ext{o}:
       - Convert to radians:    heta=315extoimesracextπrad180exto=rac7extπ4heta = 315^ ext{o} imes rac{ ext{π rad}}{180^ ext{o}} = rac{7 ext{π}}{4}    - Compute area:    A=rac12imes(142)imesrac7extπ4=rac196extπ8=24.5extπextsquarecmA = rac{1}{2} imes (14^2) imes rac{7 ext{π}}{4} = rac{196 ext{π}}{8} = 24.5 ext{π} ext{ square cm}

  5. Example 5: Give the central angle and area for an area of 240°.    - Convert to radians:
       heta=240extoimesracextπrad180exto=rac4extπ3heta = 240^ ext{o} imes rac{ ext{π rad}}{180^ ext{o}} = rac{4 ext{π}}{3}

  6. Example 6: To find areas for various conditions:    - Overall, compute specific values and relationships based on the radius and angle given, performing conversions where necessary.

Summary

  • Utilize the formulas provided for arc length and area of a sector, ensuring proper units and conversions between degrees and radians. Analyzing each example through substitution helps solidify the concepts.Possible extensions into related sectors, circles, and lengths can be explored for practice.

Exercises (Arc Length and Area of a Sector)

  • Find the length of each arc:

  1. extArcLengthfor3π/2extandradius8ext{Arc Length for } 3π / 2 ext{ and radius } 8

  2. extArcLengthfor5π/4extandradius6kmext{Arc Length for } 5π / 4 ext{ and radius } 6km

  3. Different combinations involving angles and radius units can be calculated similarly.

  • Find the area of each sector:

  1. 225exto,r=4m;225^ ext{o}, r = 4m; Calculate relevant area.

  2. Analyze sector area given non-unitary degrees and radius calculations.

  3. Engage with full circle measures where total dependency on the angle can alter the result significantly.

Conversions

Convert each degree measure to radians and vice-versa:

  • For 300exto300^ ext{o}: 300extoo300^ ext{o} o rac{5 ext{π}}{3} ext{ radians}</p></li><li><p>For</p></li><li><p>For rac{ ext{π}}{6},convertbacktodegrees:, convert back to degrees:30^ ext{o}$$; continue with given functional examples and calculations to clarify understanding through application.