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6.7 – Circumference and Arc Length

Key Terms

  • Arc Length – The length of an arc of a circle.
  • Radian – A unit of angular measure determined by the condition: The central angle of one radian in a circle of radius 1 produces an arc of length 1.

 


Review

  • Diameter
    • The diameter of a circle equals 2 radii
    • d = 2 r

 

  • Pi
    • Symbol:  π
    • π’s approximation:

GeoB 6.7 Pi

 


Notes

  • Circumference
    • Describes the distance around a circle
    • Measured in units (units, feet, inches, cm, m, etc.)
    • As the radius gets bigger, the circumference gets bigger.
    • The circumference is about 3 times the radius.

GeoB 6.7 Circumference

  • Circumference Formula
    • C = 2 π r
      • 2 times Pi times the radius
    • C = π d
      • Pi times the diameter
      • Remember: a diameter equals 2 radii

 

  • Arcs
    • Part of a circle
    • Can be measured in degrees and in units (length)

 

  • Formula to Find the Measure of an Arc length (in units)
    • Divide the arc’s degree measure by 360, then multiply by the circumference of the circle.
    • \widehat{Arc \: Length}= \frac{Arc \: Measure}{360} \cdot Circumference
    • See examples 6 – 8 (below)

GeoB 6.7 Arc Length Formula 2

GeoB 6.7 Arc Length Formula Detail

  • Radians
    • The radian is a constant of proportionality.
    • For any circle, the central angle measure (in radians) describes the ratio (fraction) between the radius and the arc length
    • There are π (pi) radians in a half-circle
    • There are 2 π (pi) radians in a full circle
      • This is why the circumference is measured with 2 π r

GeoB 6.7 Radians

GeoB 6.7 Radians Ex

 


Examples

  • Ex 1. What is the approximate circumference of the circle shown below?
    • Setup: C = 2 π r
    • Substitute:  2(3.14)(9)
    • Answer: 56.52 cm

GeoB 6.7 Ex 01

 

  • Ex 2. What is the approximate circumference of the circle shown below?
    • Setup: π d, so (3.14)(15.5)
    • Answer: 48.7 cm

GeoB 6.7 Ex 02

  • Ex 3. If you know the circumference of a circle, which step(s) can you follow to find its diameter?
    • Setup: C = π d, so you need to isolate the d.
      • Answer: Divide by π on both sides to isolate the d
      • It would look like: \frac{c}{\pi}=d

 

  • Ex 4. The radius of a circular park is 107 m. To the nearest meter, what is the circumference of the park?
    • Setup: C =  2 π r
    • Substitute: C = 2(3.14)(107)
      • Answer: 672 m

 

  • Ex 5. A blu-ray disk is shaped like a circle with a diameter of 12 cm. To the nearest centimeter, what is the circumference of the disk?
    • Setup: C = π d
    • Substitute: C = (3.14)(12)
      • Answer: 38cm

 

  • Ex 6. The circumference of \odot{F} is 72 cm. What is the length of \widehat{DE} (the minor arc)?
    • Setup: \frac{90}{360} \cdot 72
    • Simplify: \frac{1}{4} \cdot 72
      • Answer: 18 cm

GeoB 6.7 q2 Ex 01

 

  • Ex 7. The length of \widehat{DE} (the minor arc) is 22 cm. What is the circumference of \odot{F}?
    • Setup: 22=\frac{15}{360} \cdot C
    • Simplify: 22=\frac{1}{24} \cdot C
    • Multiply both sides by 24: (24)22=C
      • Answer: 528 cm

GeoB 6.7 q2 Ex 02

 

  • Ex 8. In the diagram below, what is the approximate length of the minor arc \widehat{DE}?
    • What’s missing? You need to find the circumference! Use C = 2 π r
    • C = 2(3.14)(33)
    • So, C = 207.345 cm
      • Setup: Arc=\frac{60}{360} \cdot 207.345
      • Simplify: Arc=\frac{1}{6} \cdot 207.345
      • Answer: 34.6 cm

GeoB 6.7 q2 Ex 03


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