Finding Arc Length with Degrees

Find the arc length of a sector with a central angle of $60°$ and a radius of $9$ cm.

Identify the given values: $\theta = 60°$, $r = 9$ cm = Values identified

Write the arc length formula: $s = \frac{\theta}{360°} \times 2\pi r$ = Formula ready

Substitute the values: $s = \frac{60°}{360°} \times 2\pi \times 9$ = Values substituted

Simplify the fraction: $s = \frac{1}{6} \times 18\pi$ = $\frac{60}{360} = \frac{1}{6}$

Calculate the result: $s = 3\pi \approx 9.42$ cm = $3\pi$ cm

Answer: The arc length is $3\pi$ cm, or approximately $9.42$ cm.

Finding Arc Length with Radians

A circle has a radius of $8$ m. Find the arc length for a central angle of $\frac{\pi}{4}$ radians.

Identify the given values: $\theta = \frac{\pi}{4}$ rad, $r = 8$ m = Values identified

Write the radian formula: $s = \theta \times r$ = Simpler formula for radians

Substitute and calculate: $s = \frac{\pi}{4} \times 8 = 2\pi$ = $2\pi$ m

Approximate if needed: $s = 2\pi \approx 6.28$ m = About $6.28$ m

Answer: The arc length is $2\pi$ m, or approximately $6.28$ m.

Finding the Central Angle

An arc has length $10\pi$ cm and the circle has radius $15$ cm. Find the central angle in degrees.

Identify the given values: $s = 10\pi$ cm, $r = 15$ cm = Values identified

Rearrange the formula to find the angle: $\theta = \frac{s \times 360°}{2\pi r}$ = Formula rearranged

Substitute the values: $\theta = \frac{10\pi \times 360}{2\pi \times 15}$ = Values substituted

Simplify: $\theta = \frac{3600\pi}{30\pi} = 120$ = $120°$

Answer: The central angle is $120°$.

Mistake: Using the degree formula when the angle is in radians

Why: The formulas are different! $s = \frac{\theta}{360°} \times 2\pi r$ only works for degrees.

Correct: For radians, use the simpler formula: $s = \theta \times r$. Always check the units of the angle first.

Mistake: Forgetting to convert the angle to the correct unit

Why: Mixing degrees and radians gives incorrect answers.

Correct: If given degrees but need radians: multiply by $\frac{\pi}{180}$. If given radians but need degrees: multiply by $\frac{180}{\pi}$.

Mistake: Confusing arc length with sector area

Why: Arc length is a distance (linear), while sector area is in square units.

Correct: Arc length formula: $s = \frac{\theta}{360°} \times 2\pi r$. Sector area formula: $A = \frac{\theta}{360°} \times \pi r^2$.

Clock Hands

The tip of a clock's minute hand traces an arc as time passes.

If a minute hand is $10$ cm long, it travels $2\pi \times 10 = 20\pi \approx 62.8$ cm in one full hour.

Curved Running Track

The curved portions of a running track are arcs that athletes must run.

A $400$ m track has two semicircular ends. Each semicircle is half of a full circle, so runners travel $\pi \times r$ on each curved section.

Q: Why is the radian formula simpler?

A: Radians are defined using arc length! One radian is the angle where the arc length equals the radius. So $s = \theta r$ follows directly from this definition.

Q: How do I convert between degrees and radians?

A: To convert degrees to radians: multiply by $\frac{\pi}{180}$. To convert radians to degrees: multiply by $\frac{180}{\pi}$. For example, $90° = 90 \times \frac{\pi}{180} = \frac{\pi}{2}$ radians.

Q: What is the arc length of a full circle?

A: A full circle has an angle of $360°$ (or $2\pi$ radians). The arc length is the entire circumference: $s = 2\pi r$.