To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. Both describe the same angle: a full turn is 360° or 2π radians. Decimal degrees are still degrees; degrees, minutes and seconds are another way to write them.
Start by identifying the unit attached to your number. A rotation of 1.5° is much smaller than 1.5 radians, and 12.30° does not mean 12° 30′. Keeping the unit beside each step prevents these two common mix-ups.
What is a radian?
Picture a circle with two straight lines running from its center to its edge. The angle between them can be described by the curved length between their endpoints. When that arc is as long as the circle’s radius, the angle is one radian. NIST’s SI reference gives this geometric definition.
The entire circumference is 2π times the radius, so a complete circle contains 2π radians. Half a turn is π radians, and a quarter turn is π/2 radians. Changing the circle’s size does not change those angles because the arc and radius grow together.
Convert degrees to radians
Radians = degrees × π ÷ 180
For an on-screen shape rotated through 135°, calculate 135 × π ÷ 180 = 3π/4 radians. That is the exact answer. Using a calculator’s π key gives approximately 2.356194 radians. Keep the fraction with π when you want an exact expression, or use the decimal when an input field requires a number.
For a decimal angle such as 22.5°, the process is unchanged: 22.5 × π ÷ 180 = π/8 radians, approximately 0.392699 radians. The decimal point does not introduce a different unit or require a separate conversion method.
The factor follows directly from NIST SP 330, Table 8, which defines one degree as π/180 radians. In a calculator, entering the angle × π ÷ 180 needs ordinary multiplication and division, not a sine or cosine button.
Convert radians back to degrees
Degrees = radians × 180 ÷ π
If the angle is written as 5π/6 radians, multiply by 180/π. The π terms cancel, leaving 5 × 180 ÷ 6 = 150°. For 1.2 radians, calculate 1.2 × 180 ÷ π ≈ 68.754935°. Do not multiply a decimal radian value by π again before using the formula.
A useful size check is that one radian is approximately 57.295780°. Converting a positive angle from radians to degrees therefore produces a larger number. Going from degrees to radians produces a smaller number. The physical angle stays the same.
Common angles at a glance
The exact column retains π; the decimal column is rounded to six places. These are alternative representations of each angle, not different degrees of rotation.
| Degrees | Exact radians | Decimal radians |
|---|---|---|
| 0° | 0 | 0.000000 |
| 30° | π/6 | 0.523599 |
| 45° | π/4 | 0.785398 |
| 60° | π/3 | 1.047198 |
| 90° | π/2 | 1.570796 |
| 120° | 2π/3 | 2.094395 |
| 135° | 3π/4 | 2.356194 |
| 180° | π | 3.141593 |
| 270° | 3π/2 | 4.712389 |
| 360° | 2π | 6.283185 |
Convert degrees, minutes and seconds to decimal degrees
In angular notation, one degree contains 60 minutes and one minute contains 60 seconds. The symbols are °, ′ and ″. These are subdivisions of an angle, not elapsed time. With nonnegative components, use:
Decimal degrees = degrees + minutes/60 + seconds/3600
For 18° 24′ 36″, calculate 18 + 24/60 + 36/3600 = 18.41°. Dividing both the minutes and seconds by 60 would be wrong: seconds need the factor 3600 because there are 60 × 60 angular seconds per degree.
To turn that result into radians, continue with 18.41 × π ÷ 180 ≈ 0.321315 radians. Keep the full decimal-degree value until the final step rather than rounding it before conversion.
Convert decimal degrees to degrees, minutes and seconds
Keep the whole degrees, multiply the fractional part by 60, then keep the whole minutes. Multiply the remaining fraction of a minute by 60 to obtain seconds. For 27.345°, the whole degrees are 27. The fraction 0.345 × 60 gives 20.7 minutes; the remaining 0.7 × 60 gives 42 seconds. The result is 27° 20′ 42″.
Round seconds only after splitting the components. If rounding produces 60 seconds, carry one minute and write zero seconds. If that brings the minutes to 60, carry one degree too. For example, 12° 59′ 59.8″ rounded to the nearest second becomes 13° 0′ 0″.
Keep a negative sign attached to the whole angle
For a signed angle, convert the positive magnitude first and then restore the sign. Write a negative DMS angle explicitly as −(18° 24′ 36″), giving −(18 + 24/60 + 36/3600) = −18.41°. Adding positive minutes and seconds to −18 would incorrectly give −17.59°.
This also handles values between −1° and 0°: −0.5° is −(0° 30′ 0″). Do not lose the minus sign merely because the whole-degree component is zero. Degree-to-radian conversion preserves it too: −90° equals −π/2 radians.
A geometric convention often treats counterclockwise rotation as positive, but an app may define its rotation direction differently. Use the convention of the display you are working with. Conversion alone does not change direction or remove complete turns: 450° converts to 5π/2 radians, not π/2.
When should a calculator use DEG or RAD?
Choose DEG when evaluating a trigonometric function of an angle supplied in degrees, or RAD when the angle is supplied in radians. For example, sin(30) in DEG mode and sin(π/6) in RAD mode both equal 0.5. The unit-conversion multiplications above do not depend on this setting.
Switching modes should not be assumed to convert a number already on the screen. Either use an explicit conversion feature or apply the formula and label the result. Our Desmos scientific calculator guide explains expression entry and its angle setting.
Use the π key rather than 3.14, retain extra digits during calculations, and round the final answer to suit the original measurement. If you prefer a handheld tool, browse scientific calculators on Amazon and look for clearly accessible degree and radian modes.
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