Arc Length
Learn how to calculate the length of an arc using the central angle and radius.
Definition
- = arc length
- = central angle (in degrees)
- = radius of the circle
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Worked Examples
Find the arc length of a sector with a central angle of and a radius of cm.
Identify the given values
, cm → Values identified
Write the arc length formula
→ Formula ready
Substitute the values
→ Values substituted
Simplify the fraction
→
Calculate the result
cm → cm
Answer: The arc length is cm, or approximately cm.
Common Mistakes
Using the degree formula when the angle is in radians
Why it's wrong: The formulas are different! only works for degrees.
Correct: For radians, use the simpler formula: . Always check the units of the angle first.
Forgetting to convert the angle to the correct unit
Why it's wrong: Mixing degrees and radians gives incorrect answers.
Correct: If given degrees but need radians: multiply by . If given radians but need degrees: multiply by .
Confusing arc length with sector area
Why it's wrong: Arc length is a distance (linear), while sector area is in square units.
Correct: Arc length formula: . Sector area formula: .
Interactive Visual
Unit Circle
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Interactive Sandbox
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Practice Problems
15 problemsWhat is the arc length of a full circle with radius cm?
Why It Matters
- Navigation: Pilots and sailors calculate distances along curved paths on Earth's surface
- Engineering: Designing curved roads, bridges, and roller coasters requires precise arc length calculations
- Sports: The path a ball travels when thrown in a curve, or the distance a runner covers on a curved track
- Clock hands: Finding how far the tip of a clock hand travels in a given time
Real World Applications
Clock Hands
The tip of a clock's minute hand traces an arc as time passes.
Example:
If a minute hand is cm long, it travels cm in one full hour.
A clock has a minute hand that is cm long. How far does the tip travel in minutes?
What is the arc length?
Step 1: Write the mathematical expression
In 15 minutes, the hand moves . Calculate:
Curved Running Track
The curved portions of a running track are arcs that athletes must run.
Example:
A m track has two semicircular ends. Each semicircle is half of a full circle, so runners travel on each curved section.
A running track has semicircular ends with a radius of m. How far does a runner travel on one curved end?
Find the length of one semicircular section.
Step 1: Write the mathematical expression
A semicircle is . Calculate:
Key Takeaways
- 1Arc length is the distance along the curved portion of a circle
- 2For angles in degrees:
- 3For angles in radians:
- 4The arc length is a fraction of the full circumference, proportional to the central angle
- 5Always check whether the angle is in degrees or radians before choosing the formula
Frequently Asked Questions
Glossary
- Arc
- A continuous portion of a circle's circumference
- Arc length
- The distance measured along an arc
- Central angle
- The angle formed at the center of a circle by two radii
- Radian
- A unit of angle where the arc length equals the radius (approximately )
- Sector
- The region enclosed by two radii and an arc