# 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:

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.

• 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)

• 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

Examples

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

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

• 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)

• 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)

• 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: $\frac{1}{4} \cdot 72$

• 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: $(24)22=C$
• Ex 8. In the diagram below, what is the approximate length of the minor arc $\widehat{DE}$?
• Setup: $Arc=\frac{60}{360} \cdot 207.345$
• Simplify: $Arc=\frac{1}{6} \cdot 207.345$