Arc Length and Sector Area

13 min read

A pizza is cut into a wedge, a clock hand sweeps part of a circle, a satellite travels along a curved stretch of orbit, and a sprinkler waters a pie-shaped patch of lawn. Each of these is a slice of a circle, and two questions come up again and again: how long is the curved edge, and how much area does the slice cover. Both answers fall out of one number, the central angle, the instant that angle is written in radians. The curved edge has length $s = r\theta$, and the wedge has area $A = \frac{1}{2}r^2\theta$. No integration, no geometry construction, just a radius, an angle in radians, and one multiplication. The whole trick is that radian measure was built so that the arc and the angle are the same length on a unit circle, which is exactly what makes these two formulas this short.

By the end of this page you can:

• Compute the length of a circular arc with $s = r\theta$ when the central angle is in radians.
• Compute the area of a circular sector with $A = \frac{1}{2}r^2\theta$ when the central angle is in radians.
• Convert a central angle from degrees to radians before applying either formula.
• Solve either formula for the missing quantity (find $r$, find $\theta$, or find $s$ or $A$) when the other two are given.
• Tell which formula a word problem is asking for, and keep the one-half attached to the sector-area formula.

The bare minimum: arc length is $s = r\theta$ and sector area is $A = \frac{1}{2}r^2\theta$, and in both formulas $\theta$ must be in radians. If an angle arrives in degrees, convert it to radians first. Hold onto those three facts and most problems are one substitution away from done.

Quick Reference

Before You Start

Check each box you can do from memory. A box you cannot check yet points you to a quick refresher, not a grade.

• I can convert an angle from degrees to radians by multiplying by $\pi/180$. (Angles and Radian Measure)
• I can simplify a product such as $8 \cdot \frac{\pi}{3}$ and leave the answer in terms of $\pi$.
• I can solve a one-step equation such as $24 = 6\theta$ for $\theta$.

A 60-second self-check. Pick an answer, then reveal the reasoning.

Try This First

A circle has radius $r = 4$. A central angle of $\theta = 2$ radians opens up a wedge.

Before reading further, guess two numbers: how long is the curved outer edge of the wedge, and is the wedge area bigger or smaller than the curved edge length.

The Idea Behind Arc Length and Sector Area

Radian measure is the key that makes both formulas short. An angle of $\theta$ radians is defined as the length of the arc it cuts off on a circle of radius $1$. On that unit circle the arc length and the angle are literally the same number. Scaling the radius up to $r$ scales every length on the circle by $r$, so the arc length on a circle of radius $r$ becomes $s = r\theta$. That is the entire derivation: the arc is the unit-circle arc stretched by a factor of $r$.

The sector area follows from a proportion. A full circle of radius $r$ has area $\pi r^2$ and a full turn is $2\pi$ radians. A sector with central angle $\theta$ is the fraction $\dfrac{\theta}{2\pi}$ of the whole circle, so its area is $$A = \frac{\theta}{2\pi} \cdot \pi r^2 = \frac{1}{2}r^2\theta.$$ The $\frac{1}{2}$ is not decoration. It is the $\pi$ in the area canceling against the $2\pi$ in a full turn, leaving a factor of one-half. Drop it and the answer is exactly twice too big.

Both formulas demand that $\theta$ be in radians, because both were derived from the radian definition. A degree value plugged in directly gives a number with no correct meaning. Convert first.

Prerequisite Hub

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Builds on:

Unlocks (what this section feeds into next):

Cross-course prerequisites: none.

Official Definitions

Length of a Circular Arc

In a circle of radius $r$, the length $s$ of an arc that subtends a central angle of $\theta$ radians is $$s = r\theta.$$

Stewart 7e, Section 6.1 Angle Measure (Length of a Circular Arc).

Solving this relation for $\theta$ gives a second useful form, $\theta = \dfrac{s}{r}$, which recovers the radian definition itself: the angle in radians equals the arc length divided by the radius.

Area of a Circular Sector

In a circle of radius $r$, the area $A$ of a sector with a central angle of $\theta$ radians is $$A = \tfrac{1}{2} r^2 \theta.$$

Stewart 7e, Section 6.1 Angle Measure (Area of a Circular Sector).

In both statements $\theta$ is in radians. A sector is the pie-shaped region bounded by two radii and the arc between them.

Worked Examples

Worked Example 1: Arc Length, Angle Already in Radians

Problem. Find the length of an arc that subtends a central angle of $\theta = \dfrac{\pi}{3}$ radians in a circle of radius $r = 12$.

Predict first. A full circle of radius $12$ has circumference $2\pi(12) = 24\pi \approx 75.4$. The angle $\frac{\pi}{3}$ is one-sixth of a full turn ($2\pi$), so the arc should be about one-sixth of $75.4$, roughly $12.6$. Hold that estimate.

Step 1. The angle is already in radians, so apply $s = r\theta$ directly: $$s = r\theta = 12 \cdot \frac{\pi}{3}.$$

Step 2. Simplify: $$s = \frac{12\pi}{3} = 4\pi.$$

Step 3. As a decimal, $4\pi \approx 12.57$.

Check the prediction. The estimate was about $12.6$, and $4\pi \approx 12.57$ matches. The arc length is $4\pi$, or about $12.57$ units.

Worked Example 2: Arc Length When the Angle Is in Degrees

Problem. Find the length of an arc that subtends a central angle of $30^\circ$ in a circle of radius $r = 6$.

Predict first. The angle $30^\circ$ is one-twelfth of a full turn. The circumference is $2\pi(6) = 12\pi \approx 37.7$, so the arc should be about $37.7/12 \approx 3.1$. Hold that estimate.

Step 1. The angle is in degrees, so it cannot go straight into $s = r\theta$. Convert to radians by multiplying by $\frac{\pi}{180}$: $$\theta = 30 \cdot \frac{\pi}{180} = \frac{30\pi}{180} = \frac{\pi}{6} \text{ radians}.$$

Step 2. Apply the arc-length formula: $$s = r\theta = 6 \cdot \frac{\pi}{6} = \pi.$$

Step 3. As a decimal, $\pi \approx 3.14$.

Check the prediction. The estimate was about $3.1$, and $\pi \approx 3.14$ matches. The arc length is $\pi$, or about $3.14$ units. Skipping the conversion and writing $s = 6 \cdot 30 = 180$ would have been wrong by a factor of more than fifty.

Worked Example 3: Sector Area, Keeping the One-Half

Problem. Find the area of a sector with central angle $\theta = \dfrac{\pi}{4}$ radians in a circle of radius $r = 8$.

Predict first. The full circle has area $\pi r^2 = \pi(8)^2 = 64\pi \approx 201$. The angle $\frac{\pi}{4}$ is one-eighth of a full turn, so the sector should be about $201/8 \approx 25$. Hold that estimate.

Step 1. The angle is in radians, so apply the sector-area formula and keep the one-half: $$A = \frac{1}{2}r^2\theta = \frac{1}{2}(8)^2\left(\frac{\pi}{4}\right).$$

Step 2. Compute the radius squared, then multiply: $$A = \frac{1}{2}(64)\left(\frac{\pi}{4}\right) = 32 \cdot \frac{\pi}{4} = 8\pi.$$

Step 3. As a decimal, $8\pi \approx 25.13$.

Check the prediction. The estimate was about $25$, and $8\pi \approx 25.13$ matches. The sector area is $8\pi$, or about $25.13$ square units. Forgetting the one-half would have produced $16\pi \approx 50.3$, exactly twice the correct area.

Worked Example 4: Solving for a Missing Quantity

Problem. A central angle $\theta$ in a circle of radius $r = 5$ is subtended by an arc of length $s = 20$. Find $\theta$ in radians. Then find the area of the corresponding sector.

Predict first. The arc $20$ is much longer than the radius $5$, so the angle in radians (which is arc divided by radius) should be larger than $1$, around $4$. Hold that estimate.

Step 1. Start from $s = r\theta$ and solve for $\theta$: $$\theta = \frac{s}{r} = \frac{20}{5} = 4 \text{ radians}.$$

Step 2. Now find the sector area with $r = 5$ and $\theta = 4$: $$A = \frac{1}{2}r^2\theta = \frac{1}{2}(5)^2(4) = \frac{1}{2}(25)(4) = 50.$$

Step 3. No $\pi$ appears because $\theta = 4$ is a plain real number of radians, not a multiple of $\pi$.

Check the prediction. The angle came out to $4$ radians, matching the estimate. The area is $50$ square units. A quick sanity check: $4$ radians is about $229^\circ$, more than half a full turn, so a large sector area is reasonable.

Common Misconceptions

Practice Problems

Go Deeper (optional)

None of this is needed to compute an arc length or a sector area or to pass the section. It is here for readers who want to see where these two formulas lead.

The same relation describes motion along a curve. Differentiate $s = r\theta$ with respect to time and the result is $v = r\omega$, where $v$ is the linear speed of a point moving along the arc and $\omega$ is the angular speed (how fast the angle turns, in radians per second). This is why a point on the rim of a large wheel moves faster than a point near the hub at the same rotation rate: same $\omega$, larger $r$, larger $v$. Gears, turntables, and the read head of a spinning disk all run on this one equation.

Why a careers reader might care.

• Engineering and robotics: a wheel of radius $r$ rolling without slipping advances a distance $s = r\theta$ for every $\theta$ radians it turns, which is how a robot tracks its own position by counting wheel rotations.
• Manufacturing and CNC machining: a cutting tool sweeping a circular path covers arc length $s = r\theta$, and feed rates are set from it.
• Astronomy and GPS: small-angle arc length on a large sphere (a planet, the sky) is computed with $s = r\theta$, which is how an angular separation in the sky converts to a distance.

Check Yourself

Close the notes and answer each from memory, then reveal it. Pulling an idea back from memory is one of the strongest ways to make it stick.

Mastery Checklist

Novice:

• I can write both formulas from memory: $s = r\theta$ for arc length and $A = \frac{1}{2}r^2\theta$ for sector area.
• I can state that $\theta$ must be in radians in both formulas.
• I can compute an arc length and a sector area when the angle is already in radians.

Competent:

• I can convert a degree angle to radians and then apply either formula.
• I can keep the one-half attached to the sector-area formula and explain why it is there.
• I can tell from a word problem whether it asks for arc length or sector area.

Proficient:

• I can solve either formula for a missing quantity ($r$, $\theta$, $s$, or $A$).
• I can chain the two formulas in one problem and convert the angle only once.
• I can spot a dropped one-half or a skipped degree-to-radian conversion by checking the answer against a fraction-of-the-circle estimate.

Mental Model

The slice-of-pie view. Picture a whole pizza of radius $r$. Its crust is the full circumference $2\pi r$ and its surface is the full area $\pi r^2$. Cut out one wedge with opening angle $\theta$ radians. The wedge takes the fraction $\frac{\theta}{2\pi}$ of the whole pie: its curved crust is that fraction of the circumference, which simplifies to $s = r\theta$, and its surface is that fraction of the area, which simplifies to $A = \frac{1}{2}r^2\theta$. Radians are the units that make these fractions cancel cleanly, which is the whole reason the formulas are this short.

Connections

Looking back:

• Angles and Radian Measure defines the radian as the arc length on a unit circle, which is exactly the fact that makes $s = r\theta$ work.

Looking ahead:

• Linear and angular speed extend $s = r\theta$ over time into $v = r\omega$, relating how fast a point travels along the arc to how fast the angle turns.

Real-world connections:

• A rolling wheel of radius $r$ advances $s = r\theta$ per $\theta$ radians turned, used in odometry and robotics.
• A sprinkler or radar sweep waters or scans a sector of area $A = \frac{1}{2}r^2\theta$.
• Gear and pulley systems transmit motion through the arc-length relation between meshed rims.

Resources

Last updated: 2026-06-24