Trigonometry guide

Radians vs Degrees

Q: Is pi radians the same as pi degrees?

No. The unit is part of the quantity. An angle of π rad is exactly 180^°, while π^° is only about 3.1416^°. The symbol π is a number, not an angle unit by itself. Keep the degree symbol or radian label attached when comparing quantities, and check whether a converter expects radians or just the coefficient of pi radians.

Q: Do all radian values need to contain pi?

No. An angle can be exactly 1 rad, 2 rad or any other real radian value. Rational multiples of pi are convenient for many familiar degree angles, so they appear frequently in tables. A terminating decimal entered as an exact radian value usually converts to an irrational degree value. Keep its unit explicit and distinguish an exact input from a rounded measurement.

Q: Can I use degrees in the arc-length formula?

Convert the angle first when using s=rθ, because that compact formula assumes the numerical angle is in radians. For a nonnegative central degree measure d, the equivalent formula is s=π rd/180. Use matching length units for the radius and result. For a signed sweep, ordinary traveled distance uses the magnitude of the angle, while a signed arc displacement can preserve its sign.

Two units, the same rotation. Learn what radians measure, convert angles exactly, read the unit circle and choose the right calculator mode. Worked examples keep direction, completed turns and rounding clear.

Common-angle reference · Four geometric illustrations · Practice with answers

Open degrees and radians converter

What is the difference between radians and degrees?

A degree divides one complete turn into $360$ equal parts. A radian measures a central angle through the circle itself: divide the intercepted arc length $s$ by the radius $r$ , using the same length unit for both. The ratio gives the numerical radian measure $\theta$ . The radius must be positive.

\theta=\frac{s}{r}\quad\text{(numerical radian measure)}

One radian

\theta=1\,\mathrm{rad}

s=r

One radian is defined by the curved arc matching the radius, not by a particular circle size.

One radian comes from equal lengths

When the curved arc length equals the radius, $s=r$ , so the angle is $1\,\mathrm{rad}$ . This is about $57.2958^\circ$ . A larger circle has a proportionally longer arc for the same angle; the ratio stays unchanged. The straight chord connecting the endpoints is shorter than the curved arc and does not define radian measure.

A circle's circumference is $2\pi r$ . Dividing that full-turn arc by its radius gives $2\pi$ . A half turn is therefore $\pi\,\mathrm{rad}$ , and a quarter turn is $\pi/2\,\mathrm{rad}$ . These relationships are exact.

360^\circ=2\pi\,\mathrm{rad}

180^\circ=\pi\,\mathrm{rad}

The BIPM SI Brochure identifies the radian as the coherent unit for plane angle. Although a ratio of lengths has unit one, writing the radian symbol makes the quantity clear. Degrees remain useful for navigation, construction, geometry and everyday angle measurements.

A circle with a counterclockwise quarter-turn from the right axis to the top axis, beside exact degree and radian equivalents for zero through one full turn. — The highlighted quarter-turn runs counterclockwise from the right axis to the top axis. The table keeps zero and a complete turn as different rotation amounts.

How to convert degrees and radians

Let $d$ be an angle's numerical measure in degrees and $t$ its numerical measure in radians. The half-turn relationship gives two conversion rules. Use the first for degrees to radians and the second for radians to degrees; the destination unit belongs on the final angle.

Degrees to radians

t=d\frac{\pi}{180}

Radians to degrees

d=t\frac{180}{\pi}

Example one: convert an obtuse degree angle

Start with $135^\circ$ . Multiply its numerical degree measure by $\pi/180$ . $t=135\frac{\pi}{180}=\frac{135\pi}{180}$
Reduce the fraction by dividing its numerator and denominator by $45$ . Leave pi exact. $135^\circ=\frac{3\pi}{4}\,\mathrm{rad}\approx2.3562\,\mathrm{rad}$
Check the size: the input lies between $90^\circ$ and $180^\circ$ , so the output lies between $\pi/2$ and $\pi$ radians. The decimal is an approximation; the fraction is exact.

Check the degree-to-radian example

Example two: convert a pi-based radian angle

Start with $7\pi/6\,\mathrm{rad}$ . Multiply the numerical radian measure by $180/\pi$ . $d=\frac{7\pi}{6}\frac{180}{\pi}$
Cancel the common factor of pi, then divide $180$ by $6$ and multiply by $7$ . $d=7\times30=210$
Attach the degree symbol. The exact result is $210^\circ$ , a little more than a half turn. Do not enter a pi coefficient as an ordinary radian value without also including pi.

Check the radian-to-degree example

Example three: a radian value without pi

Radians do not have to be written as a multiple of pi. For an exact input of $1\,\mathrm{rad}$ , apply the same rule. Here pi remains in the denominator, so the exact degree value is not a terminating decimal.

1\,\mathrm{rad}=\frac{180}{\pi}{}^\circ\approx57.2958^\circ

Retain the exact expression until the last step. If the input came from a physical measurement, its uncertainty still matters; extra displayed digits do not improve that measurement.

Convert one radian to degrees

Common degrees and radians table

Use the exact column when simplifying formulas. The decimal column rounds only the final radian value to four places. Each degree entry opens that conversion in the calculator, including the original full-turn amount.

Swipe across the table if needed. Keyboard users can focus the table and use the arrow keys.

One complete turn: exact angles and rounded equivalents
Degrees	Radians, exact	Radians, rounded
$0^\circ$	$0\,\mathrm{rad}$	$0.0000\,\mathrm{rad}$
$30^\circ$	$\frac{\pi}{6}\,\mathrm{rad}$	$0.5236\,\mathrm{rad}$
$45^\circ$	$\frac{\pi}{4}\,\mathrm{rad}$	$0.7854\,\mathrm{rad}$
$60^\circ$	$\frac{\pi}{3}\,\mathrm{rad}$	$1.0472\,\mathrm{rad}$
$90^\circ$	$\frac{\pi}{2}\,\mathrm{rad}$	$1.5708\,\mathrm{rad}$
$120^\circ$	$\frac{2\pi}{3}\,\mathrm{rad}$	$2.0944\,\mathrm{rad}$
$135^\circ$	$\frac{3\pi}{4}\,\mathrm{rad}$	$2.3562\,\mathrm{rad}$
$150^\circ$	$\frac{5\pi}{6}\,\mathrm{rad}$	$2.6180\,\mathrm{rad}$
$180^\circ$	$\pi\,\mathrm{rad}$	$3.1416\,\mathrm{rad}$
$210^\circ$	$\frac{7\pi}{6}\,\mathrm{rad}$	$3.6652\,\mathrm{rad}$
$225^\circ$	$\frac{5\pi}{4}\,\mathrm{rad}$	$3.9270\,\mathrm{rad}$
$240^\circ$	$\frac{4\pi}{3}\,\mathrm{rad}$	$4.1888\,\mathrm{rad}$
$270^\circ$	$\frac{3\pi}{2}\,\mathrm{rad}$	$4.7124\,\mathrm{rad}$
$300^\circ$	$\frac{5\pi}{3}\,\mathrm{rad}$	$5.2360\,\mathrm{rad}$
$315^\circ$	$\frac{7\pi}{4}\,\mathrm{rad}$	$5.4978\,\mathrm{rad}$
$330^\circ$	$\frac{11\pi}{6}\,\mathrm{rad}$	$5.7596\,\mathrm{rad}$
$360^\circ$	$2\pi\,\mathrm{rad}$	$6.2832\,\mathrm{rad}$

Read the same angle on the unit circle

A second-quadrant angle

\theta=135^\circ=\frac{3\pi}{4}\,\mathrm{rad}

(x,y)=\left(-\frac{\sqrt2}{2},\frac{\sqrt2}{2}\right)

The point is left of the vertical axis and above the horizontal axis. Both axes use the same scale.

A unit circle has radius $1$ and center $(0,0)$ . In standard position, the initial ray points along the positive horizontal axis. Positive angles turn counterclockwise. The terminal point is $(\cos\theta,\sin\theta)$ when $\theta$ is expressed in the intended angle convention.

For $135^\circ=3\pi/4\,\mathrm{rad}$ , the terminal ray lies in the second quadrant. Its reference angle is $45^\circ$ , so both coordinate magnitudes are $\sqrt2/2$ . The horizontal coordinate is negative and the vertical coordinate is positive.

\cos\left(\frac{3\pi}{4}\right)=-\frac{\sqrt2}{2}

\sin\left(\frac{3\pi}{4}\right)=\frac{\sqrt2}{2}

In these formulas, the trigonometric arguments use radians. The diagram's signs provide a quick check on the answers. OpenStax explains the coordinate definitions and exact special-angle values.

Explore this angle on the unit circle

Negative angles, extra turns and coterminal angles

A negative angle turns clockwise from the initial ray. Conversion preserves the sign. Adding a complete turn changes the rotation amount but returns the terminal ray to the same direction; such angles are called coterminal.

Same terminal ray, different rotations

\text{Dashed: }-45^\circ

\text{Solid: }315^\circ

Arrowheads show the direction of rotation. The two arc guides use different radii for clarity; compare their angles, not their drawn lengths.

A negative quarter of a right angle

-45^\circ=-\frac{\pi}{4}\,\mathrm{rad}

To choose a coterminal representative in $[0^\circ,360^\circ)$ , add one complete turn. The square bracket includes zero; the round bracket excludes the full-turn endpoint.

-45^\circ+360^\circ=315^\circ

315^\circ=\frac{7\pi}{4}\,\mathrm{rad}

These are the same terminal direction, but different signed rotations. The calculator displays the original conversion and an optional wrapped value separately. A conversion alone must not replace a negative angle with its positive coterminal representative.

Compare the negative angle and its wrapped value

Keep completed turns when they matter

An angle of $450^\circ$ is one full turn followed by a quarter turn. Converting it gives $5\pi/2\,\mathrm{rad}$ . Wrapping it to a single positive turn gives $90^\circ$ , or $\pi/2\,\mathrm{rad}$ , but discards the completed rotation. Keep the original angle for wheel travel, rotation counts and signed displacements.

450^\circ=360^\circ+90^\circ=\frac{5\pi}{2}\,\mathrm{rad}

Likewise, $360^\circ$ converts to $2\pi\,\mathrm{rad}$ , not zero. For any integer $k$ , radian angles $\theta$ and $\theta+2\pi k$ are coterminal. Their sine and cosine agree even though the signed rotation amounts may differ.

Keep the extra quarter-turn

Convert degrees, minutes and seconds first

An arcminute is $1/60$ of a degree, and an arcsecond is $1/3600$ of a degree. These are angular units, not clock time. For nonnegative degree, arcminute and arcsecond components $D$ , $M$ and $S$ , form the decimal degree magnitude before converting it.

d=D+\frac{M}{60}+\frac{S}{3600}

For $12^\circ30'0''$ , divide the arcminutes by $60$ to get half a degree.
$d=12+\frac{30}{60}=12.5$
$12.5^\circ=12.5\frac{\pi}{180}\,\mathrm{rad}=\frac{5\pi}{72}\,\mathrm{rad}$

For a negative compound angle, apply the sign to the whole magnitude: $-(12+30/60)=-12.5$ , not $-12+30/60=-11.5$ . The converter has a separate sign control to make this interpretation explicit. In normalized components, arcminutes and arcseconds each stay below $60$ .

Check the degrees-and-minutes example Check the whole-angle negative sign

Avoid DEG and RAD calculator mistakes

The same typed number means different angles in different modes. In degree mode, entering $30$ into sine means $30^\circ$ . In radian mode, it means $30\,\mathrm{rad}$ , which is several complete turns. The outputs should be different.

\sin(30^\circ)=\frac12

\sin(30\,\mathrm{rad})\approx-0.9880316

Read the unit supplied by the problem before entering a number.
Select DEG for degrees or RAD for radians. In the Unit Circle Calculator, use the explicit input-angle convention.
Evaluate the function, then check its sign against the terminal ray's quadrant. Do not relabel the result to repair a mode mismatch.

Compare the degree-mode result Compare the radian-mode result

Use radians for arc length and sector area

Let $r$ be radius and $\theta$ the numerical central-angle measure in radians. For a nonnegative sweep, arc length is $s=r\theta$ . For a simple sector with $0\leq\theta\leq2\pi$ , area is $A=\frac12r^2\theta$ . Radius and arc length use matching length units; sector area uses their squared unit.

Arc and sector

r=6\,\mathrm{cm}

\theta=60^\circ=\frac{\pi}{3}\,\mathrm{rad}

The gold region is one sixth of the disk. The curved boundary and the filled region measure different quantities.

Example: a sixth of a disk

Given $r=6\,\mathrm{cm}$ and a central angle of $60^\circ$ , convert first: $60^\circ=\pi/3\,\mathrm{rad}$ .
Multiply the radius by the numerical radian measure.
$s=6\frac{\pi}{3}=2\pi\,\mathrm{cm}$ $s\approx6.2832\,\mathrm{cm}$
For the filled sector, square the radius and include the half factor.
$A=\frac12(6)^2\frac{\pi}{3}=6\pi\,\mathrm{cm}^2$ $A\approx18.8496\,\mathrm{cm}^2$

Check: the full circumference is $12\pi\,\mathrm{cm}$ and full disk area is $36\pi\,\mathrm{cm}^2$ . Both answers are one sixth of their full-circle quantities.

Inserting the degree number $60$ directly into $s=r\theta$ would give the wrong answer. For a signed sweep, ordinary traveled arc distance uses $s=r|\theta|$ . A signed arc displacement can retain the sign. Repeated turns add travel distance but do not create a larger distinct sector inside the same disk.

See the Circle Calculator for circle dimensions and Area Formulas for square units and composite regions. The arc and sector relationships are also derived in OpenStax's angle chapter.

Why radians appear in calculus

Standard trigonometric derivatives assume radian input. For example, $\frac{d}{dx}\sin x=\cos x$ uses a numerical radian argument $x$ . If a function instead accepts a numerical degree input $d$ , write the conversion inside it. The chain rule introduces a scale factor.

f(d)=\sin\left(\frac{\pi d}{180}\right)

f'(d)=\frac{\pi}{180}\cos\left(\frac{\pi d}{180}\right)

This is why the unit convention matters beyond calculator buttons. OpenStax's trigonometric derivative lesson gives the standard formulas.

Try three quick conversions

Simplify exactly before checking a decimal. Open each answer with a click, or focus its heading and press Enter or Space.

Convert

225^\circ

to radians.

Multiply by $\pi/180$ , then reduce: $225\pi/180=5\pi/4$ . The answer is $5\pi/4\,\mathrm{rad}\approx3.9270\,\mathrm{rad}$ . It lies in the third quadrant, between a half turn and three quarters of a turn.

Convert

-5\pi/3\,\mathrm{rad}

to degrees.

Multiply by $180/\pi$ : $(-5\pi/3)(180/\pi)=-300$ . The original angle is $-300^\circ$ . Its positive coterminal representative is $60^\circ$ , but that wrapped value is a separate answer to a different question.

Find the arc length for

r=4\,\mathrm{m}

and a nonnegative sweep of

1.5\,\mathrm{rad}

The angle is already in radians, so $s=r\theta=4(1.5)=6\,\mathrm{m}$ . Do not multiply by $\pi/180$ again. The output is a length, not an area, so its unit is meters rather than square meters.

FAQ

Frequently Asked Questions

What is the difference between radians and degrees?

They measure the same kind of quantity using different scales. A full rotation is

360^\circ

in degrees and

2\pi\,\mathrm{rad}

in radians. Radian measure comes from the ratio of circular arc length to radius, with both lengths in matching units. Converting changes the numerical representation, not the geometric rotation. Neither system makes an imprecise measurement more accurate just by changing its label.

Is pi radians the same as pi degrees?

No. The unit is part of the quantity. An angle of

\pi\,\mathrm{rad}

is exactly

180^\circ

, while

\pi^\circ

is only about

3.1416^\circ

. The symbol

\pi

is a number, not an angle unit by itself. Keep the degree symbol or radian label attached when comparing quantities, and check whether a converter expects radians or just the coefficient of pi radians.

Do all radian values need to contain pi?

No. An angle can be exactly

1\,\mathrm{rad}

2\,\mathrm{rad}

or any other real radian value. Rational multiples of pi are convenient for many familiar degree angles, so they appear frequently in tables. A terminating decimal entered as an exact radian value usually converts to an irrational degree value. Keep its unit explicit and distinguish an exact input from a rounded measurement.

Why does a negative angle become positive when I normalize it?

Normalization chooses a representative in a stated range, usually

[0^\circ,360^\circ)

. It does not change the original converted angle. For example,

-45^\circ

converts to

-\pi/4\,\mathrm{rad}

; its positive coterminal representative is

315^\circ

. The two rotations end on the same ray but describe different signed amounts of rotation. Retain the original value whenever direction or completed turns matter.

Are zero degrees and a full turn equal?

They have the same terminal direction in standard position, but they are different rotation amounts:

0^\circ

is no turn and

360^\circ

is one complete turn. The latter converts to

2\pi\,\mathrm{rad}

, not zero. A wrapping operation can map both to the same representative, and sine and cosine agree for them, but a travel-distance or rotation-count calculation must preserve the completed turn.

Which calculator mode should I use for sine and cosine?

Match the mode to the angle you are entering. Use DEG for a degree value and RAD for a radian value, or use a tool with an explicit input-unit selector. The identity

\sin(30^\circ)=1/2

is a useful check; the sine of thirty radians is different. Change the mode before evaluating the function rather than relabeling an already computed output.

Can I use degrees in the arc-length formula?

Convert the angle first when using

s=r\theta

, because that compact formula assumes the numerical angle is in radians. For a nonnegative central degree measure

d

, the equivalent formula is

s=\pi rd/180

. Use matching length units for the radius and result. For a signed sweep, ordinary traveled distance uses the magnitude of the angle, while a signed arc displacement can preserve its sign.

When should I round a degree-radian conversion?

Keep pi and rational factors exact while simplifying, then round once at the final decimal answer. Replacing pi with a short decimal too early introduces avoidable error, especially across several calculations. The converter keeps a symbolic expression alongside a rounded view. If the original angle was measured approximately, an exact conversion relationship still does not imply that every displayed decimal digit is physically meaningful.

Sources and calculation checks

The definitions follow the primary references below. The worked conversions, coordinate signs, signed arc directions, sector example and table decimals were independently checked using exact symbolic arithmetic. The calculator links preserve each example's input and angle convention. Approximation signs identify rounded outputs.

OpenStax: Angles
Degree and radian measure, standard position, coterminal angles, arc length and sectors.
OpenStax: Unit Circle, Sine and Cosine
Exact special-angle coordinates and the connection between circular position and trigonometric functions.
BIPM: The International System of Units
The current SI Brochure identifies the radian as the coherent unit for plane angle; see sections 2.3.3 and 2.3.4 and the accepted non-SI units table.
OpenStax: Derivatives of Trigonometric Functions
Standard trigonometric derivative formulas; the guide's degree-input comparison follows by the chain rule.

Continue with related tools and guides

Degrees and Radians Converter
Convert exact pi expressions, compound angles and signed rotations.
Unit Circle Calculator
Check coordinates, quadrants and exact trigonometric values.
Right Triangle Calculator
Apply angle relationships to triangle sides and acute angles.
Polar Graph Paper
Plot angles on a circular grid with degree or radian labels.
Math Formulas
Look up related geometry and trigonometry relationships.
Slope and Intercept Explained
Connect coordinate directions with lines and their gradients.

Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. The method was last reviewed on October 29, 2026. No independent professional review is claimed.

Methodology

The editorial team checks definitions, examples, linked tools, and cited references. Dates shown here come from the page's maintained publication record.

Read the editorial methodology

Dates and sources

Published June 6, 2026. Updated October 29, 2026.