Degrees vs. Radians in Arc Length: Formulas, Conversion, and Examples

One angle, two units, and the wrong one can change your answer.

Why can the same angle be 90 degrees and π/2 radians? Both measure the turn of a central angle, but they count it differently: degrees split a full turn into 360 equal parts, while radians measure the turn in radius lengths. The unit decides which arc-length formula works, and mixing them up is one of the most common errors in geometry. This guide explains both systems, shows how to convert between them, and works one arc-length problem both ways. Check any result with our Arc Length Calculator, and read Arc Length of a Circle for the full circle formulas.

What Are Degrees and Radians?

Degrees split a circle into 360 parts; radians measure it in radius lengths.

Understanding Degrees

A complete circle contains 360 degrees, a half circle 180 degrees, and a quarter circle 90 degrees. A right angle is 90°, a straight line is 180°, and a full turn brings you back to 360°. Degrees suit everyday measurement, because 360 divides neatly into halves, quarters, thirds, and sixths.

Understanding Radians

A radian measures an angle by the arc it cuts off.

One radian is the central angle that subtends an arc equal in length to the radius. That definition links angle, arc length, and radius:

$\theta = \frac{L}{r}$

Here $\theta$ is the angle in radians, $L$ the arc length, and $r$ the radius. If the radius is 5 cm and the arc is 5 cm long, the angle is 1 radian. A full circumference is $2\pi r$ , so a full circle is $2\pi$ radians and a half circle is $\pi$ radians. That gives the key link between the two systems:

$180^\circ = \pi \text{ radians}$

A full circle showing 360 degrees equals 2 pi radians, with 90 degrees and 180 degrees marked

What Is the Difference Between Degrees and Radians?

Same angle, different units, like inches and centimetres.

Degrees and radians describe the same turn, only the numbers differ. A quarter turn is $90^\circ$ or $\frac{\pi}{2}$ radians, a half turn is $180^\circ$ or $\pi$ radians, and a full turn is $360^\circ$ or $2\pi$ radians. Neither unit is more correct; the right choice depends on the problem.

Feature	Degrees	Radians
Full circle	$360^\circ$	$2\pi$ radians
Half circle	$180^\circ$	$\pi$ radians
Quarter circle	$90^\circ$	$\frac{\pi}{2}$ radians
Measurement system	360 equal parts of a turn	Arc length in radius lengths
Common applications	School geometry, navigation, maps	Trigonometry, calculus, physics
Arc-length formula	$L = \frac{\theta}{360^\circ} \times 2\pi r$	$L = r\theta$

How to Convert Degrees into Radians

Multiply by π/180, then simplify the fraction.

The conversion formula is:

$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

It works because $180^\circ = \pi$ radians, so one degree equals $\frac{\pi}{180}$ radians.

Example 1: Convert 60 degrees into radians.

$60 \times \frac{\pi}{180}$ .
Simplify: $\frac{60}{180} = \frac{1}{3}$ .
$\frac{\pi}{3}$ radians.

Example 2: Convert 135 degrees into radians.

$135 \times \frac{\pi}{180}$ .
Simplify: $\frac{135}{180} = \frac{3}{4}$ .
$\frac{3\pi}{4}$ radians.

These are the angles that appear most often, and they are worth learning by heart.

Degrees	Radians
$0^\circ$	$0$
$30^\circ$	$\frac{\pi}{6}$
$45^\circ$	$\frac{\pi}{4}$
$60^\circ$	$\frac{\pi}{3}$
$90^\circ$	$\frac{\pi}{2}$
$120^\circ$	$\frac{2\pi}{3}$
$135^\circ$	$\frac{3\pi}{4}$
$150^\circ$	$\frac{5\pi}{6}$
$180^\circ$	$\pi$
$270^\circ$	$\frac{3\pi}{2}$
$360^\circ$	$2\pi$

Two semicircles comparing degree values 0, 90 and 180 with radian values 0, pi over 2 and pi

How to Convert Radians into Degrees

Multiply by 180/π, the reverse of the first conversion.

The conversion formula is:

$\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

It works because $\pi$ radians $= 180^\circ$ , so one radian equals $\frac{180}{\pi}$ degrees, about $57.3^\circ$ .

Example 1: Convert $\frac{2\pi}{3}$ radians into degrees.

$\frac{2\pi}{3} \times \frac{180}{\pi}$ .
Cancel $\pi$ : $\frac{2 \times 180}{3}$ .
$120^\circ$ .

Example 2: Convert $\frac{5\pi}{6}$ radians into degrees.

$\frac{5\pi}{6} \times \frac{180}{\pi}$ .
Cancel $\pi$ : $\frac{5 \times 180}{6} = 5 \times 30$ .
$150^\circ$ .

Some angles have no tidy fraction of $\pi$ , so the answer is a decimal: 1.2 radians is $1.2 \times \frac{180}{\pi} \approx 68.75^\circ$ . That is the difference between an exact answer and an approximate one. Keep $\pi$ in the working until the last step, and round only at the end.

How Do Degrees and Radians Affect Arc Length?

The unit picks the formula, and the radian formula is the shorter one.

Arc length using degrees:

$L = \frac{\theta}{360^\circ} \times 2\pi r$

It takes the right fraction of the circumference: $90^\circ$ is $\frac{1}{4}$ of the circle, so the arc is a quarter of $2\pi r$ .

Arc length using radians:

$L = r\theta$

Here the angle must be in radians. It comes from $\theta = \frac{L}{r}$ : multiply both sides by $r$ to get $L = r\theta$ . Radians make the formula simpler because a radian already measures the angle in radius lengths.

Method	Formula	Use it when
Degrees	$L = \frac{\theta}{360^\circ} \times 2\pi r$	The angle is given in degrees
Radians	$L = r\theta$	The angle is given in radians

Move the slider below and watch both formulas give the same length every time.

Adjust Central Angle 90° (1.57 rad)

Radius is fixed at 10 for this comparison.

Degree Formula

L = 2πr × (θ / 360)

Calculation: 2 × π × 10 × (90 / 360)

Result 15.71

Radian Formula

L = r × θ

Calculation: 10 × 1.571

Result 15.71

Both formulas always yield the exact same Arc Length!

How to Calculate the Same Arc Length Using Both Methods

Two routes, one answer: 2π, about 6.28 cm.

A circle has a radius of 8 cm and a central angle of 45 degrees. Calculate the arc length.

Method 1: Degrees

$L = \frac{45}{360} \times 2\pi \times 8$ .
Simplify: $\frac{45}{360} = \frac{1}{8}$ .
Circumference: $2\pi \times 8 = 16\pi$ .
Multiply: $L = \frac{1}{8} \times 16\pi = 2\pi \approx 6.28$ cm.

Method 2: Radians

Convert first: $45^\circ = \frac{\pi}{4}$ radians.
$L = 8 \times \frac{\pi}{4}$ .
Multiply: $L = 2\pi \approx 6.28$ cm.

Both methods give the same result because $45^\circ$ and $\frac{\pi}{4}$ radians are the same angle in different units. Pick the method that matches the unit you were given, and convert only when you need to.

A 45 degree sector with radius 8 centimetres showing the arc length is 2 pi, about 6.28 centimetres

When Should You Use Degrees or Radians?

Use the unit the problem gives you, and convert only when the formula asks for the other.

Use the degree formula when the angle is in degrees and the problem needs no trigonometry, as in most school geometry problems.
Use the radian formula when the angle is already in radians: one multiplication, the faster route.
Convert first when the angle and the formula disagree: degrees going into $L = r\theta$ must become radians first.
Use radians in advanced mathematics: trigonometry, calculus, and physics all work in radians.
Use degrees in basic geometry: maps, navigation, and everyday measurement still run on degrees.

In short, check the unit on the angle, pick the formula that matches it, and convert only when a later step needs the other unit.

Calculator Guidance

Mode matters for trig keys, not for plain multiplication.

Most scientific calculators have a degree mode and a radian mode. That setting only changes trigonometric and inverse-trigonometric keys: $\sin$ , $\cos$ , $\tan$ , and their inverses. It does not change ordinary multiplication, so $L = r\theta$ needs a radian angle regardless of mode. If your angle is $45^\circ$ and you type $8 \times 45$ , the calculator multiplies by 45, not by $\frac{\pi}{4}$ , and the answer is wrong in either mode. Convert $45^\circ$ to $\frac{\pi}{4}$ first, or use the degree formula instead. Check the mode before any problem that uses sine or cosine.

Common Mistakes When Using Degrees and Radians

Seven slips, each with a quick fix.

Using degrees directly in $L = r\theta$ . An angle of $90^\circ$ is not 90 radians: with $r = 10$ cm, $10 \times 90 = 900$ cm is wrong, while $10 \times \frac{\pi}{2} \approx 15.71$ cm is right.
Using the wrong conversion factor. Degrees to radians multiplies by $\frac{\pi}{180}$ , not $\frac{180}{\pi}$ . Swapping them turns $90^\circ$ into about 5,157 radians.
Confusing 180 degrees with 180 radians. 180 radians is about 10,313 degrees, more than 28 full turns, so always check the unit after the number.
Forgetting that 360 degrees equals 2π radians. A full circle is $2\pi$ radians, not $\pi$ : using $\pi$ for a full turn halves the answer.
Rounding too early. Keep $\pi$ in the working and round only the final answer.
Using the wrong calculator mode for trigonometric operations. Sine and cosine answers depend on the mode, so confirm DEG or RAD first.
Forgetting to label angle measurements. Write $^\circ$ for degrees and “rad” for radians: an unlabelled 1.57 could be either.

Practice Problems: Degrees, Radians, and Arc Length

Five questions, then check your work below.

Convert 90 degrees into radians.
Convert 150 degrees into radians.
Convert $\frac{3\pi}{4}$ radians into degrees.
A circle has a radius of 10 cm and a central angle of 60 degrees. Find its arc length using both formulas.
A circle has a radius of 5 cm and a central angle of $\frac{\pi}{3}$ radians. Calculate its arc length.

Answer key:

$90 \times \frac{\pi}{180} = \frac{\pi}{2}$ radians.
$150 \times \frac{\pi}{180} = \frac{5\pi}{6}$ radians.
$\frac{3\pi}{4} \times \frac{180}{\pi} = 135^\circ$ .
Degrees: $L = \frac{60}{360} \times 2\pi \times 10 = \frac{10\pi}{3} \approx 10.47$ cm. Radians: $60^\circ = \frac{\pi}{3}$ , so $L = 10 \times \frac{\pi}{3} = \frac{10\pi}{3} \approx 10.47$ cm.
$L = r\theta = 5 \times \frac{\pi}{3} = \frac{5\pi}{3} \approx 5.24$ cm.

Frequently Asked Questions

Short answers.

1. What is the difference between degrees and radians? Both measure the same angle in different units: degrees split a full turn into 360 parts, radians measure it in radius lengths, and a full turn is $2\pi$ radians.

2. How many radians are in 360 degrees? $2\pi$ radians, about 6.28 radians.

3. How do you convert degrees into radians? Multiply the degrees by $\frac{\pi}{180}$ . For example, $90^\circ \times \frac{\pi}{180} = \frac{\pi}{2}$ radians.

4. How do you convert radians into degrees? Multiply the radians by $\frac{180}{\pi}$ . For example, $\frac{\pi}{2} \times \frac{180}{\pi} = 90^\circ$ .

5. Why does the arc-length formula use radians? A radian is defined by $\theta = \frac{L}{r}$ , which rearranges to the short form $L = r\theta$ . Radians measure the angle in radius lengths, so no 360 is needed.

6. Can you calculate arc length using degrees? Yes: use $L = \frac{\theta}{360^\circ} \times 2\pi r$ , which takes the right fraction of the full circumference.

7. What happens if you use degrees in $L = r\theta$ ? The answer comes out far too large: with $r = 10$ cm and $90^\circ$ , $10 \times 90 = 900$ cm, while the correct arc is about 15.71 cm. Convert to radians first.

8. Which is easier for calculating arc length? Radians, when the angle is already in radians: $L = r\theta$ is one multiplication. With degrees, use the degree formula and convert only when needed.

9. Is one radian equal to 57.3 degrees? Yes, about $57.3^\circ$ : exactly $\frac{180}{\pi}$ degrees, about $57.2958^\circ$ .

10. Should a calculator be in degree or radian mode? Match the mode to the angle’s unit, and only for trig keys: for $L = r\theta$ the mode does not matter, since the angle must be in radians either way.