Choose the angle required by the formula
When using an arc-length formula, convert the angle to radians and keep the radius in the desired length unit. The result’s length unit then follows from the radius.
The following inputs stay in degrees and the results use radians. They illustrate different magnitudes for this measurement. Retain the specific conditions described above when replacing an example with your own value.
| Starting value | Equivalent value |
|---|---|
| 30 ° | 0.523598775598 rad |
| 45 ° | 0.785398163397 rad |
| 90 ° | 1.57079632679 rad |
This angle unit pair has a constant factor ratio. Changing the input changes the output by the same proportion. If a copied figure differs by an unexpected factor, review the full unit symbol before changing the source measurement. Signed angles can describe direction. The converter preserves the sign rather than normalizing it.
Convert the angle to radians, retain the radius in the chosen length unit, and state the reference direction if the sign or orientation matters.
Choose radians before multiplying by the radius
A drawing labels an angle in degrees, but an arc-length formula expects radians. The number 45 is not ready for that equation until its unit is converted. This small mismatch can produce a result that looks numerical and plausible while describing the wrong arc. Keep the radius and angle units explicit before using the formula.
For a 0.4 m radius and a 45° angle, first convert the angle to π/4 radians, approximately 0.785398163397. Arc length is then 0.4 × π/4 = about 0.314159265359 m. Multiplying the radius directly by 45 would produce 18 m, which is not the intended arc. A simple bound helps: a 45° arc is one eighth of the full circumference.
Chord length is a different quantity. For the same circle it is 2r sin(θ/2), approximately 0.306146745892 m. The straight chord and curved arc should not be substituted merely because both are lengths associated with the same endpoints. A diagram clarifies which one the task needs.
Most programming trigonometric functions expect radians, but calculator modes and spreadsheet functions should be checked individually. Signed rotations, bearings measured clockwise from north and mathematical angles measured counterclockwise from an axis need their orientation conventions preserved. A unit conversion changes degrees into radians without choosing a starting direction or automatically normalizing a negative angle to the range 0–360°.
A 45° arc at radius 0.4 m has length π/10 m.
45 ° in radians
45 ° = 0.785398163397 rad
Output = input × source factor ÷ destination factor45 × 0.0174532925199 ÷ 1 = 0.785398163397 rad. The two factors express the source and destination relative to the same internal reference unit. Their ratio is the multiplier.
Both unit factors refer to the same internal reference for angle. Multiplying by the source factor and dividing by the destination factor changes the numerical scale while retaining the quantity. The source definition still matters: Convert degrees to radians first; the unit conversion is distinct from choosing a reference direction.
Reversing the unrounded 0.785398163397 rad result returns 45 °. Use this as a direction check. A consistent reverse calculation cannot resolve passing a degree value to a radians-only function or improve the original observation.
Angle units, side by side
The values below all describe 45 °. They provide a comparison of unit sizes for angle; they are not separate measurements. Read the complete symbol and convention before copying a number.
- °Degrees
- 45 °
- radRadians
- 0.785398163397 rad
- gradGradians
- 50 grad
- turnTurns
- 0.125 turn
Keep the original degrees record alongside the converted radians display. The unit list uses up to twelve significant digits for arithmetic readability, while the measurement precision still comes from its source.
Angles need a zero direction
An angle describes rotation or the separation of directions. Degrees, radians, gradians and turns are useful in drawings, trigonometry and rotational work. The converter retains the supplied signed angle; it does not automatically wrap it into a compass heading or decide which coordinate axis the angle was measured from.
One full turn is 360 degrees, 400 gradians or 2π radians. Radians connect geometry directly to arc length: s = rθ uses θ in radians. π has no finite exact decimal representation, so a displayed radian equivalent is rounded even when a degree-to-turn ratio is exact. A bearing convention is a separate coordinate choice.
For a circular arc, s = rθ uses θ in radians. Degrees connect to radians through π/180, and a full turn is 2π radians. Converting the angle does not establish the centre of the circle or the start direction. A signed angle and a compass bearing also have different conventions; normalization to a single turn is a separate operation.
Convert the angle to radians, retain the radius in the chosen length unit, and state the reference direction if the sign or orientation matters.
Questions about angle
Why does 180 degrees produce a rounded decimal?
Its exact radian equivalent is π. The calculator displays a finite approximation because JavaScript uses floating-point numbers and the output is limited to twelve significant digits.
Will 450 degrees automatically become 90 degrees?
No. The converter changes units without normalizing rotations. Four hundred fifty degrees is 1.25 turns; reducing it modulo one turn is a separate operation.
Before using this angle result
- Check whether the application expects degrees or radians before using a trigonometric function.
- Keep the reference direction and clockwise/counterclockwise convention with the value.
- Do not confuse decimal degrees with degrees-minutes-seconds notation.
Keep the original angle value, its unit and the relevant conditions in the record. A copied answer of 0.785398163397 rad is interpretable only when those details remain available. State which of the measurement checks above you verified before using the value in another calculation.
What this angle comparison can tell you
Useful capabilities
Convert the angle to radians, retain the radius in the chosen length unit, and state the reference direction if the sign or orientation matters.
The listed units express the same supplied angle quantity. The 45 ° → 0.785398163397 rad example makes the selected convention visible before it is used in a longer calculation.
Important limits
Radians use π, so a finite decimal display is rounded. Signed angles are supported.
Convert degrees to radians first; the unit conversion is distinct from choosing a reference direction. Signed angles can describe direction. The converter preserves the sign rather than normalizing it.
Passing a degree value to a radians-only function
Convert degrees to radians first; the unit conversion is distinct from choosing a reference direction.
Assuming a negative angle is invalid
Signed angles can describe direction. The converter preserves the sign rather than normalizing it.
Angular measurement accuracy and orientation conventions are different issues. A degree-to-radian factor can be exact while the measured angle remains uncertain. A bearing may be clockwise from north, while a mathematical angle is counterclockwise from an axis. Preserve that convention and do not apply automatic wrapping when a signed rotation is meaningful. Arc length and chord length also need distinct labels.
As a controlled uncertainty example, suppose the original 45 ° record had a stated ±1% input interval. Its endpoints would be 44.55 ° and 45.45 °. Converting those endpoints gives an output interval from 0.777544181763 rad to 0.793252145031 rad. The interval scales with the same fixed unit ratio as the central value. This chosen interval illustrates the arithmetic and is not an accuracy claim for your measurement.