Arc Length

Learn how to calculate the length of an arc using the central angle and radius.

Intermediate25 minLesson

Definition

An arc is a portion of a circle's circumference. The arc length is the distance along the curved line of the arc.

To find arc length, we use the formula:

s = \frac{\theta}{360°} \times 2\pi r

where:

$s$ = arc length
$\theta$ = central angle (in degrees)
$r$ = radius of the circle

Alternatively, if the angle is in radians:

s = \theta \times r

This simpler formula works because radians are defined using arc length!

Try it now

What is the arc length of a full circle with radius

5

cm?

Worked Examples

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.

Common Mistakes

Using the degree formula when the angle is in radians

Why it's wrong: 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.

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

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

Interactive Visual

Unit Circle

Degrees

0°

Radians

sin(θ)

cos(θ)

tan(θ)

Coordinates (cos, sin)

(1, 0)

Click and drag to rotate the angle around the unit circle.

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems

Problem 1 of 15

Easy

What is the arc length of a full circle with radius $5$ cm?

Why It Matters

Arc length appears everywhere in the real world:

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

Understanding arc length connects geometry to real-world measurements and prepares you for more advanced topics like calculus.

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

1Try It Yourself

A clock has a minute hand that is $12$ cm long. How far does the tip travel in $15$ minutes?

What is the arc length?

Step 1: Write the mathematical expression

In 15 minutes, the hand moves $90°$ . Calculate: $\frac{90}{360} \times 2\pi \times 12$

Curved Running Track

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

Example:

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.

2Try It Yourself

A running track has semicircular ends with a radius of $35$ 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 $180°$ . Calculate: $\frac{180}{360} \times 2\pi \times 35$

Key Takeaways

1Arc length is the distance along the curved portion of a circle
2For angles in degrees: $s = \frac{\theta}{360°} \times 2\pi r$
3For angles in radians: $s = \theta \times r$
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

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.

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 $57.3°$ )
Sector: The region enclosed by two radii and an arc