Angle Measure: Radians and Degrees

15 min read

Set a calculator to compute $\sin(30)$. Many students expect $0.5$, the value of $\sin 30^\circ$, and instead read $-0.988$. Nothing is broken. The calculator measured the angle in radians, not degrees, and $30$ radians is a completely different angle than $30$ degrees. Degrees and radians are two units for the same thing, the size of a turn, the way feet and meters are two units for length. A trig course runs almost entirely in radians, so the first job is to move between the two units without losing track of which one is in front of you.

By the end of this page you can:

• State what one radian is, in terms of the arc cut off by a central angle and the radius of the circle.
• Convert a degree measure to radians by multiplying by $\dfrac{\pi}{180^\circ}$, and a radian measure to degrees by multiplying by $\dfrac{180^\circ}{\pi}$.
• Read an angle written with no unit symbol as a radian measure, and explain why a radian carries no unit.
• Find an angle coterminal with a given angle by adding or subtracting a full turn ($360^\circ$ or $2\pi$).
• Tell a coterminal angle apart from a reference angle.

The floor to take away: one half turn is $\pi$ radians, and $\pi$ radians is $180^\circ$. From that single equation, every degree-radian conversion is one multiplication, and a coterminal angle is one addition or subtraction of a full turn. A student who holds only $\pi \text{ radians} = 180^\circ$ can rebuild the rest.

Quick Reference

Before You Start

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

• I can sketch a central angle and see the arc it cuts off along the circle. (Angles in Degree Measure)
• I can place an angle in standard position, vertex at the origin, initial side along the positive $x$-axis.
• I know that a full turn is $360^\circ$ and a half turn is $180^\circ$.

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

Try This First

A pizza has a radius of $6$ inches. Lay a piece of string along the curved crust, starting at one straight cut. Stop the string after it has covered a length of exactly $6$ inches of crust, the same as the radius. Now draw the two radii that reach the ends of that piece of crust.

Before reading on, predict: is the angle at the center of the pizza bigger or smaller than a right angle ($90^\circ$)?

The Idea Behind Radian Measure

Degrees split a full turn into $360$ equal parts. That number $360$ comes from history, not from the circle. There is a second way to measure an angle that comes from the circle directly.

As a central angle opens wider, the arc it cuts off along the circle grows longer. The arc length and the angle change together: a wider angle traces a longer arc, a narrower angle traces a shorter one. Radian measure reads the size of the angle straight off the length of that arc, in units of the radius.

Picture the radius as a measuring stick. Lay it along the arc. If the arc is exactly one radius long, the angle is one radian. If the arc is two radii long, the angle is two radians. The radian count is the number of radii that fit along the intercepted arc.

There are two correct ways to see a radian. The first: a radian is a fixed amount of turn, about $57.3^\circ$, that could be marked on a protractor. The second: a radian is a ratio, arc length divided by radius, so it does not depend on which circle is drawn. The ratio view is the steadier one once several circles of different sizes appear, because the radian for the small circle and the radian for the large circle name the same angle.

Prerequisite Hub

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Builds on:

Unlocks (what this section feeds into next):

Cross-course prerequisites: none.

Official Definitions

Radian

If a circle of radius $1$ has the vertex of the angle at its center, the measure of the angle in radians is the length of the arc that the angle subtends.

Stewart 7e, Section 6.1 Angle Measure. Equivalently, one radian is the measure of a central angle that subtends an arc equal in length to the radius of the circle.

The OpenStax alternate states the same idea: “One radian is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle.” (OpenStax Precalculus 2e, Section 5.1 Angles.)

Degree-radian relationship

Because the circumference of a circle of radius $1$ is $2\pi$, a full turn subtends an arc of length $2\pi$, so a full turn is $2\pi$ radians. A half turn is therefore $\pi$ radians, which is also $180^\circ$. That single fact, \[ \pi \text{ radians} = 180^\circ, \] is the bridge between the two units. From it: \[ 1 \text{ radian} = \frac{180^\circ}{\pi} \approx 57.3^\circ, \qquad 1^\circ = \frac{\pi}{180} \text{ radians}. \]

Stewart 7e, Section 6.1 Angle Measure.

Standard position and coterminal angles

An angle is in standard position if its vertex is at the origin and its initial side lies along the positive $x$-axis. Two angles are coterminal if they are both in standard position and their terminal sides coincide.

Stewart 7e, Section 6.1 Angle Measure.

Radians carry no unit

A radian is the ratio of two lengths, arc length divided by radius, so the length units cancel and a radian is a pure number. When an angle is written as $\dfrac{\pi}{2}$ or $2$ with no degree symbol, it is in radians. Writing a degree symbol on a radian measure, such as $\pi^\circ$, changes the meaning entirely and is incorrect.

How to Convert: the Two Ratios

To convert, multiply by whichever ratio cancels the unit being left.

The single trap is using the wrong ratio. The check that prevents it: degrees-to-radians should make the number smaller (a degree is a tiny angle, so a large count of them collapses to a small radian number), and radians-to-degrees should make the number larger.

The common angles in both units.

Worked Examples

Each example states a prediction first, then computes, then compares.

Example 1: Degrees to radians

Convert $135^\circ$ to radians.

Predict. A half turn ($180^\circ$) is $\pi$ radians, and $135^\circ$ is three quarters of $180^\circ$, so expect roughly $\dfrac{3}{4}\pi$, a number between $2$ and $3$.

Compute. Multiply by $\dfrac{\pi}{180^\circ}$: \[ 135^\circ \cdot \frac{\pi}{180^\circ} = \frac{135\pi}{180} = \frac{3\pi}{4} \text{ radians}. \]

Compare. The result $\dfrac{3\pi}{4} \approx 2.36$ radians sits between $2$ and $3$, and the number got smaller than $135$, both matching the prediction.

Example 2: Radians to degrees

Convert $\dfrac{5\pi}{6}$ radians to degrees.

Predict. $\pi$ radians is $180^\circ$, and $\dfrac{5\pi}{6}$ is five sixths of $\pi$, so expect five sixths of $180^\circ$, a bit less than $180^\circ$.

Compute. Multiply by $\dfrac{180^\circ}{\pi}$: \[ \frac{5\pi}{6} \cdot \frac{180^\circ}{\pi} = \frac{5 \cdot 180^\circ}{6} = \frac{900^\circ}{6} = 150^\circ. \]

Compare. $150^\circ$ is just under $180^\circ$, and the number got larger than the starting value of about $2.6$, as predicted.

Example 3: A radian is a pure number

A circle of radius $3$ centimeters has an arc $6$ centimeters long. Find the central angle in radians.

Predict. The arc is twice the radius, so two radii fit along it, so expect $2$ radians, with no unit attached.

Compute. A radian count is arc length divided by radius: \[ \theta = \frac{s}{r} = \frac{6 \text{ cm}}{3 \text{ cm}} = 2. \] The centimeters in the numerator and denominator cancel.

Compare. The angle is $2$ radians, a pure number, even though both lengths were given in centimeters. The unit canceled, which is exactly why a radian is unitless. Calling the answer “$2$ centimeters” would confuse the angle with the arc that produced it.

Example 4: Converting an unfamiliar degree angle

Convert $40^\circ$ to radians, leaving the answer in terms of $\pi$.

Predict. $40^\circ$ is between $30^\circ$ ($\tfrac{\pi}{6}$) and $45^\circ$ ($\tfrac{\pi}{4}$), so expect a radian value between $\tfrac{\pi}{6} \approx 0.52$ and $\tfrac{\pi}{4} \approx 0.79$.

Compute. Multiply by $\dfrac{\pi}{180^\circ}$ and reduce the fraction: \[ 40^\circ \cdot \frac{\pi}{180^\circ} = \frac{40\pi}{180} = \frac{2\pi}{9}. \]

Compare. Numerically $\dfrac{2\pi}{9} \approx 0.70$, which falls between $0.52$ and $0.79$, as predicted.

Coterminal Angles

Two angles in standard position are coterminal when their terminal sides land in exactly the same place. Adding a full turn to an angle returns the ray to where it started, so the result names the same terminal side. A full turn is $360^\circ$ in degrees and $2\pi$ in radians. Adding or subtracting any whole number of full turns produces a coterminal angle.

To find a coterminal angle, add or subtract $360^\circ$ (or $2\pi$) as many times as needed. To find the coterminal angle in a standard range such as $0^\circ$ to $360^\circ$ (or $0$ to $2\pi$), add or subtract full turns until the result lands in that range.

Example 5: A positive coterminal angle for a negative angle

Find an angle between $0^\circ$ and $360^\circ$ that is coterminal with $-60^\circ$.

Predict. A $-60^\circ$ angle opens $60^\circ$ clockwise from the positive $x$-axis, ending in the fourth quadrant, $60^\circ$ short of a full turn. Expect the answer to be $360^\circ - 60^\circ = 300^\circ$.

Compute. Add one full turn: \[ -60^\circ + 360^\circ = 300^\circ. \] Since $300^\circ$ is between $0^\circ$ and $360^\circ$, stop.

Compare. $300^\circ$ lands in the fourth quadrant, the same terminal side as $-60^\circ$, matching the prediction.

Example 6: Reducing a large radian angle

Find an angle between $0$ and $2\pi$ that is coterminal with $\dfrac{13\pi}{4}$.

Predict. A full turn is $2\pi = \dfrac{8\pi}{4}$. Since $\dfrac{13\pi}{4}$ is more than $2\pi$ but less than $4\pi = \dfrac{16\pi}{4}$, subtracting one full turn should land in range. Expect $\dfrac{13\pi}{4} - \dfrac{8\pi}{4} = \dfrac{5\pi}{4}$.

Compute. Subtract one full turn: \[ \frac{13\pi}{4} - 2\pi = \frac{13\pi}{4} - \frac{8\pi}{4} = \frac{5\pi}{4}. \] Since $\dfrac{5\pi}{4}$ is between $0$ and $2\pi$, stop.

Compare. $\dfrac{5\pi}{4} \approx 3.93$ falls between $0$ and $2\pi \approx 6.28$, as predicted.

Common Misconceptions

Practice Problems

Go Deeper (optional)

None of this is needed to convert an angle or to find a coterminal angle. It is here for readers who want to see why radians dominate later courses.

The reward for radians shows up the moment trig meets calculus. The derivative of $\sin x$ is $\cos x$ only when $x$ is in radians; in degrees an extra factor of $\dfrac{\pi}{180}$ appears and never goes away. Every numerical library, from a spreadsheet to a game engine to a control system, expects angles in radians by default, which is the reason a calculator left in radian mode returned $-0.988$ for $\sin(30)$ in the opener. The habit of checking the angle unit before trusting a trig output prevents one of the most common and costly bugs in scientific and graphics code.

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 (Level 1-2):

• I can state the definition of a radian in terms of arc length and radius.
• I can convert between degrees and radians using the ratios $\dfrac{\pi}{180^\circ}$ and $\dfrac{180^\circ}{\pi}$.
• I can read an angle written with no symbol as a radian measure.

Competent (Level 3-4):

• I can find a central angle from a given arc length and radius using $\theta = \dfrac{s}{r}$.
• I can find an angle coterminal with a given angle by adding or subtracting full turns.
• I can reduce a coterminal angle into a standard range such as $0$ to $2\pi$.

Proficient (Level 5):

• I can tell a coterminal angle apart from a reference angle and explain the difference.
• I can explain why a radian is a unitless ratio that does not depend on the size of the circle.
• I can explain why an angle with no unit symbol is read in radians.

Mental Model

Picture a radian as measuring an angle with the radius as the ruler. Take the radius and lay it along the curved edge like a flexible ruler. The number of radius lengths that fit along the intercepted arc is the angle in radians. Two radii fit, the angle is $2$ radians. The whole way around takes $2\pi$ radii, so a full turn is $2\pi$ radians. Coterminal angles are what happens when the ruler goes all the way around and keeps going: adding a full turn lands the terminal side back where it started. The one equation to anchor everything is $\pi \text{ radians} = 180^\circ$.

Connections

Looking back:

• Angles in Degree Measure sets up the central-angle picture and the $360^\circ$ full turn that radian measure replaces with $2\pi$.

Looking ahead:

• Arc Length and Sector Area use $s = r\theta$ and $A = \tfrac{1}{2} r^2 \theta$, formulas that are clean only when the angle is in radians.
• Reference Angles are stated in radians once radian measure is in hand, and they sit next to the coterminal idea introduced here.

Real-world connections:

• Programming languages and engineering tools expect angles in radians by default, so converting and checking the unit is a routine first step.
• A calculator silently in radian mode returns a wrong-looking trig value for a degree input, so checking the angle unit before trusting a trig output is a routine habit.

Resources

Last updated: 2026-06-24