Arc length: how far you travel around the rim

Explain it like I am five

Picture walking along the curved edge of a round garden. The distance you walk is the arc length, called s. Radians count how many radius-lengths the walk uses, so the distance is the radius times that count: s = rθ, where r is the radius and θ (theta) is the angle in radians.

Example: the garden's radius is r = 5 m (meters) and the central angle, the angle at the center, is 144° (144 degrees). The formula needs radians, so convert first: 144° × $π over 180°$ = $144 π over 180$ = $4 π over 5$ , dividing top and bottom by 36. Then s = rθ = 5 × $4 π over 5$ = 4π m, about 12.57 m.

Why it works: $4 π over 5$ radians means the walk uses $4 π over 5$ , about 2.51, radius-lengths, and each one is 5 m long. Check it another way: 144° is $144 over 360$ = $2 over 5$ of a full turn, and the whole edge is 2π × 5 = 10π m. $2 over 5$ × 10π = 4π m, the same answer.

In plain words

Imagine walking along the curved edge of a circular garden. The distance you walk is the arc length (s), while the straight distance from the center to the edge is the radius. Radians tell you how many radius-lengths your walk uses. If the turn uses three radius-lengths and each radius is 4 meters, you walk 12 meters. This also works backward: divide the walk's length by the radius to find the angle, or divide by the radian count to find the radius. For a physical slice, use the nonnegative size of its opening, because a distance along the rim cannot be negative.

The arc is of the circumference because its opening is of a revolution.

Reminder

Undoing multiplication. To solve 14 = 2r, divide both sides by 2 to leave the unknown radius alone: r = 7. Check 2 × 7 = 14.
Dividing by a fraction. 10π ÷ $2 π over 5$ = 10π × $5 over 2 π$ = 25. Multiplying by the reciprocal undoes the fraction divisor.
Magnitude. Magnitude means size without the turning sign. A chosen clockwise 72° arc uses a positive 72° size; do not replace it with a different arc.

Why it works. A central angle of θ radians uses of one full turn. Its arc therefore uses that same fraction of the circumference: s = × 2πr = rθ. The 2π factors cancel. This explains both the formula and its unit requirement: radians compare the angle with a 2π-radian full turn. Degrees compare with 360° instead. Dividing s = rθ by r or by θ gives the two reverse formulas without changing the relationship.

Rules = rθ; θ = ; r = when θ > 0. θ must be in radians.
For one physical slice, r > 0 and 0 ≤ θ ≤ 2π; arc length is a nonnegative size, also called its magnitude.

The same idea, five ways

Say it

Say arc length is radius times the central angle in radians.

Write it

Arc length measures the curved distance along the chosen rim; the radian opening counts how many radius-lengths it contains.

In math

s = rθ
θ = $\frac{s}{r}$ , with r > 0
r = $\frac{s}{θ}$ , with θ > 0
for a single arc: 0 ≤ θ ≤ 2π, using its nonnegative opening
θ in radians; s and r in matching length units

The radius is one ribbon piece, the radian angle is the piece count, and the arc is the combined ribbon length.

See it

The chosen arc is one fifth of this circle's circumference.

The same idea, other ways

Count measuring sticks

An angle of 2 radians tells you to lay two radius-lengths along the curved rim. If a radius-length is 7 cm, the arc uses 2 × 7 = 14 cm. Multiplication combines the number of sticks with the length of each stick.

Two radius-lengths of 7 cm make a 14 cm arc.

Keep the same fraction of the rim

A quarter turn keeps a quarter of a full circumference; a fifth turn keeps a fifth. The angle's fraction of a full turn is the arc's fraction of the full rim. In radians, multiplying $θ over 2 π$ by 2πr leaves rθ after canceling 2π.

A fifth of a revolution cuts off a fifth of the circumference.

.1Find arc length

The radian count says how many measuring sticks to lay along the rim, and the radius says how long each stick is. Multiplying the count by the stick's length gives the curved distance. A longer radius makes a longer arc for the same opening.

s = rθ with θ in radians.
Length answers use units such as centimeters or meters, not square units.
For a clockwise description of a physical slice, use the magnitude of its stated opening, not a negative distance.

A 60° opening uses one sixth of the rim.

Reminder

Degree conversion and reduction. 75° × $π over 180°$ = $75 π over 180$ = $5 π over 12$ , dividing top and bottom by 15.

Worked exampleA radian angle is ready to use

You are finding distance along the curved rim using the radius and a given radian opening. Find the arc length for radius 9 cm and central angle $π over 3$ radians.

Read the radius along the straight spoke and the arc length along the curved rim.

s = 9 × $π over 3$ cm.The angle is already in radians, so substitute directly into s = rθ.
s = 3π cm.The factor 9 divided by 3 equals 3.

Answer

s = 3π cm, exactly.

Check = 60°, which is of a full turn. One sixth of 2π × 9 = 18π cm is 3π cm.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: For radius 8 cm and angle 75°, s = 8 × 75 = 600 cm.

s = rθ counts radius-lengths, so θ must be measured in radians. The degree count 75 cannot be substituted directly.

✓ Instead: 75° = radians, so s = 8 × = cm.

Tips and tricks

Write radians beside θ before multiplying. Give the result in a length unit.

.2Find the central angle

You know how far the arc goes and how large the circle is. Divide the arc by the radius to count how many radius-lengths it contains. This produces radians even when the lengths are in centimeters. Convert afterward if the question also wants a degree measurement.

θ = $\frac{s}{r}$ is the radian definition and the reverse of s = rθ.
The length units cancel when s and r use the same unit.
Convert the resulting radian measurement to degrees with $180° over π radians$ .

The 15 cm arc contains two-and-a-half copies of its 6 cm radius.

Reminder

Radian-to-degree conversion. 2.5 radians × $180° over π radians$ = $450° over π$ ≈ 143.24°, rounded to two decimal places.

Worked exampleRecover the angle from an arc and radius

You are finding a turn from its curved distance, then expressing that turn in both units. An arc of 15 cm lies on a circle of radius 6 cm. Find the central angle in radians and in degrees to two decimal places.

Read the radius along the straight spoke and the arc length along the curved rim.

θ = $15 cm over 6 cm$ = $5 over 2$ = 2.5 radians.Divide arc by radius; the centimeters cancel, and dividing 15 and 6 by 3 gives $5 over 2$ .
θ = 2.5 radians × $180° over π radians$ = $450° over π$ .The degree conversion cancels radians and multiplies 2.5 by 180.
θ ≈ 143.24°.450 ÷ π ≈ 143.2394488, which rounds to 143.24 at two decimal places.

Answer

Exact angle: 2.5 radians.
Degree angle: approximately 143.24°.

Check The arc uses = of the 12π cm circumference. That same fraction of 360° is × 360° = , independently reproducing the degree answer.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: An arc of 15 cm with radius 6 cm gives a central angle of 2.5°.

The ratio counts radius-lengths and therefore produces radians, not degrees.

✓ Instead: θ = = 2.5 radians. Convert afterward to about 143.24° if degrees are requested.

Tips and tricks

Use the same length unit for the arc and radius, then write radians after their ratio.

.3Find the radius

If you know the arc length and the number of radius-lengths in it, divide the total distance by the count. It is like dividing a roll of ribbon among a known number of equal pieces. The result is the length of one piece, which here is the radius.

From s = rθ, divide both sides by θ to get r = $\frac{s}{θ}$ .
θ must be positive when recovering a radius from a physical arc.
A zero opening has zero arc length for every radius, so it cannot determine the radius.

Dividing the 9π cm arc by its radian count recovers a 27 cm radius.

Reminder

Dividing by a fraction. Dividing by $π over 3$ multiplies by its reciprocal $3 over π$ , whose product with $π over 3$ is 1.

Worked exampleDivide by a fraction of π

You are finding the straight distance from the center to the rim by undoing the given radian count. An arc of 9π cm subtends a central angle of $π over 3$ radians. Find the circle's radius.

Read the radius along the straight spoke and the arc length along the curved rim.

9π = r × $π over 3$ .Substitute the arc and radian angle into s = rθ.
r = 9π ÷ $π over 3$ = 9π × $3 over π$ .Divide both sides by the nonzero angle; dividing by a fraction means multiplying by its reciprocal.
r = 27 cm.π cancels and 9 × 3 = 27; the remaining unit is a length.

Answer

r = 27 cm.

Check The opening is 60°, or one sixth of a turn. Six arcs total 6 × 9π = 54π cm. The circumference of a radius-27 circle is also 2π × 27 = 54π cm.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: For s = 9π cm and θ = , find r by multiplying 9π by .

The radius was already multiplied by θ in s = rθ. Recovering the radius must undo that multiplication.

✓ Instead: r = 9π ÷ = 9π × = 27 cm.

Tips and tricks

Divide by the positive radian angle, then substitute your radius back into s = rθ.

Strategy: step by step

Identify the chosen arc, its radius and its central angle. Use the physical opening's nonnegative size, because the length measures distance rather than clockwise or counterclockwise direction. Magnitude means size with the direction sign removed: −75° has magnitude 75°. A chosen 300° arc remains 300°; using 60° would select a different arc.
If the angle is in degrees, convert it to radians first by multiplying by $π radians over 180°$ .
Choose the form that isolates the missing quantity: multiply rθ for length, divide s by r for angle, or divide s by θ for radius.
Use matching length units for s and r. Keep exact π expressions through the calculation and round only the final decimal answer.
Check independently using the angle's fraction of a full turn times the circumference. For a slice no larger than the whole circle, its arc cannot exceed 2πr.

Strategy

Select the missing arc quantity

Is the given opening in degrees?

YesConvert with before using s = rθ.

NoA radian opening is ready to use.

↓

Is arc length the unknown?

YesMultiply rθ.

NoIdentify whether the angle or the radius is unknown.

↓

Is the central angle the unknown?

YesDivide s by r; this finds the radian count.

NoFor the radius, divide s by positive θ; this finds the length of one radius-sized piece.

Identify the chosen arc, central opening, and radius from the question.
Use the nonnegative size of that chosen opening and convert degrees to radians if necessary.
Find which quantity is missing: arc length, central angle, or radius.
Use s = rθ, θ = $\frac{s}{r}$ , or r = $\frac{s}{θ}$ to leave that unknown alone.
Check matching length units and label the requested answer.
Verify with the fraction of a full turn times the circumference or by substituting the recovered quantity back.

Worked exampleConvert a degree opening before finding the arc

You are finding distance along the specified curved rim and expressing that length exactly and as a rounded decimal. A circle has radius 8 cm. Find the arc cut off by a 75° central angle, exactly and to two decimal places.

Read the radius along the straight spoke and the arc length along the curved rim.

θ = 75° × $π radians over 180°$ = $75 π over 180$ = $5 π over 12$ radians.s = rθ requires radians; dividing 75 and 180 by 15 reduces the fraction.
s = 8 × $5 π over 12$ = $40 π over 12$ cm.The radian count tells how many 8 cm radius-lengths fit along the arc.
s = $10 π over 3$ cm.Divide numerator and denominator by 4 to keep the exact result reduced.
s ≈ 10.47 cm. $10 π over 3$ ≈ 10.4719755, which rounds to 10.47 at two decimal places.

Answer

Exact arc length: $10 π over 3$ cm.
Rounded arc length: approximately 10.47 cm.

Check 75° uses = of the circle. Its circumference is 2π × 8 = 16π cm. Thus × 16π = = cm, the same result without the radian arc formula.

Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.

Rung 1Rung 1: one radius-length

You are finding distance along the curved rim, rather than straight across the circle. Find the arc length when r = 5 cm and θ = 1 radian.

Read the radius along the straight spoke and the arc length along the curved rim.

s = 5 × 1 = 5 cm.One radian uses one radius-length of arc, and the radius is 5 cm.

Answer

s = 5 cm.

Check One radian is of a full turn; that fraction of the 10π cm circumference is × 10π = 5 cm.

Rung 2Rung 2: decimal radians

You are finding distance along the curved rim, rather than straight across the circle. A circle has radius 10 units. How long is its arc for θ = 1.2 radians?

Read the radius along the straight spoke and the arc length along the curved rim.

s = 10 × 1.2.The angle is already in radians, so s = rθ applies directly.
s = 12 units.Ten copies of 1.2 total 12, and arc length uses the radius's length unit.

Answer

s = 12 units.

Check The turn fraction is . Multiply it by the 20π-unit circumference: × 20π = 12 units.

Rung 3Rung 3: degrees need a conversion

You are finding distance along the curved rim, rather than straight across the circle. Find the arc length for r = 8 cm and θ = 75°, exactly and to two decimal places.

Read the radius along the straight spoke and the arc length along the curved rim.

75° × $π radians over 180°$ = $75 π over 180$ = $5 π over 12$ radians.The arc formula needs radians; reduce by dividing 75 and 180 by 15.
s = 8 × $5 π over 12$ = $40 π over 12$ = $10 π over 3$ cm.Multiply radius by radian angle, then divide the numerator and denominator by 4.
s ≈ 10.47 cm.The exact answer is about 10.4719755 cm, which rounds to two decimal places.

Answer

Exact: $10 π over 3$ cm.
Rounded: approximately 10.47 cm.

Check The angle occupies = of a revolution. That fraction of 16π cm is cm.

Rung 4Rung 4: recover the radius

You are finding the straight distance from center to rim using the known curved length and radian opening. An arc has length 7.2 cm and central angle 0.8 radians. Find its radius.

Read the radius along the straight spoke and the arc length along the curved rim.

7.2 = 0.8r.Substitute the known arc length and radian angle into s = rθ.
r = $7.2 over 0.8$ .Divide both sides by 0.8 to undo multiplication and leave r alone.
r = $72 over 8$ = 9 cm.Multiplying top and bottom by 10 removes both decimals without changing the ratio, and 72 ÷ 8 = 9.

Answer

r = 9 cm.

Check The arc fraction is . For the proposed radius, circumference is 18π cm, so × 18π = 7.2 cm, the given arc.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: With radius 6 cm and angle 50°, calculate s = 6 × 50 = 300 cm.

The number 50 counts degrees, while s = rθ needs a count of radius-lengths in radians.

✓ Instead: Convert 50° to radians first. Then s = 6 × = cm.

✗ Not this: A clockwise opening of 72° has a negative physical arc length.

Clockwise supplies the turning sign, but physical length measures a nonnegative size.

✓ Instead: Use opening size 72° = radians for the chosen arc's length, while the signed turn may be −.

✗ Not this: A zero arc and zero opening reveal a radius through r = .

Every radius produces zero arc at zero opening. Division by zero is undefined, and the data cannot distinguish any radius.

✓ Instead: Recover a radius with r = only when the physical opening θ is positive.

Tips and tricks

Write radians beside θ before substituting into the arc formula.
Use length units for arc length, such as cm. Use square units only for area.
For one arc no longer than a complete circle, compare your answer with circumference 2πr.

Trap. Putting 75 directly into s = rθ when the opening is 75°. The formula counts radius-lengths, so θ must be radians. Write the conversion before substituting, and keep the answer in a length unit rather than degrees.

Keep in mind

θ must be in radians: putting 144 straight into s = rθ gives 5 × 144 = 720 m, far more than the whole 10π ≈ 31.4 m edge.
Run the formula backward when needed: θ = $\frac{s}{r}$ , so a 15 m walk on a 6 m radius is $15 over 6$ = 2.5 radians.
Keep π exact until the last step: 4π ≈ 12.57, but rounding π first gives 3.14 × 4 = 12.56.
A distance is never negative, so use the size of the opening: a −65° slice uses 65°.

Memory hookSay it as "s equals r theta": lay the radius along the rim θ times. Radians only, so convert degrees first.

Flash cards: say the answer out loud, then flip

What is arc length?

The distance along the curved edge of a circle between the two sides of a central angle.

State the arc length formula and its unit rule.

s = rθ, with θ in radians

r = 10 cm and θ = . Find s.

3π cm, about 9.42 cm

r = 12 cm and θ = 25°. Find s exactly.

25° = , so s = 12 × = cm, about 5.24 cm

Can the arc of one slice be longer than 2πr?

No. 2πr is the whole edge, so a longer answer means the angle was not converted to radians.