Formula

Converting between degrees and radians

Trigonometry • Trigonometric Identity • Degree and Radian

Formula

π radians = 18 0^{\circ}

π

radians

= 18 0^{\circ}

, so

x^{\circ}

\frac{π x}{180}

radians and

y

radians is

(\frac{180 y}{π})^{\circ}

; in particular

1

radian

\approx 57. 3^{\circ}

and

1^{\circ} \approx 0.0175

radians.

How to use it

1Multiply by $\frac{π}{180}$ to pass from degrees to radians and by $\frac{180}{π}$ to come back.
2Hold on to the size: one radian is nearly $6 0^{\circ}$ , so a small bare number written as an angle is a large angle, not a small one.
3The superscript $c$ , as in $1^{c}$ , marks circular measure; a trigonometric ratio of a bare number always means the radian value.

Express

13 5^{\circ}

in radians, and

\frac{π}{5}

radians in degrees.

Step-by-step

1
$13 5^{\circ} = 135 \times \frac{π}{180} = \frac{3 π}{4} radians$
Why: Multiply by $\frac{π}{180}$ .
2
$\frac{π}{5} radians = \frac{π}{5} \times \frac{180}{π} = 3 6^{\circ}$
Why: Multiply by $\frac{180}{π}$ ; the $π$ cancels.

Answer

\frac{3 π}{4}

radians and

3 6^{\circ}

Watch out

sin 1

means

sin (1^{c}) \approx 0.841

, not

sin 1^{\circ} \approx 0.017

; the two differ by a factor of nearly

50

. The arc-length and sector-area formulas

s = r θ

and

A = \frac{1}{2} r^{2} θ

hold only with

θ

in radians.

Also known as

radian measure, circular measure, degree to radian, pi radians equals 180 degrees, convert angle units