Direct answer
One complete turn equals 360 degrees, 2π radians, or 400 gradians; conversion multiplies by the appropriate ratio while preserving the same physical rotation.
What this calculation tells you
Angular units partition a complete rotation differently. Degrees are familiar in everyday geometry, radians connect angles with circle geometry and calculus, and gradians divide a turn into 400 parts.
Coterminal angles can differ by whole turns while ending in the same orientation.
Where it is used
Education and mathematics
Move between geometry, trigonometry, and calculus conventions.
Engineering
Communicate rotation and phase in the unit required by formulas or equipment.
Surveying
Work with declared angular systems without silently mixing them.
Computing and graphics
Supply angles to software APIs that often expect radians.
When this guide helps
- Converting degrees to radians.
- Reading an angle supplied in turns.
- Checking a trigonometric calculator mode.
- Comparing coterminal orientations.
Use the full-turn identity
Every conversion follows from 360°=2π radians=400 gradians=1 turn. Keep π exact until a decimal is required.
Understand radians
A radian measures arc length divided by radius, which is why circle and calculus formulas simplify in radians.
Separate orientation from accumulated rotation
An angle of one turn and zero have the same final orientation but represent different accumulated rotations in some applications.
Common mistakes
Frequent errors include entering degrees in radian mode and replacing π too early.
- Label the unit.
- Keep π symbolic when possible.
- Check expected quadrant and scale.
Worked case: half turn
Convert 180 degrees.
180/360=0.5 turn; radians=180xπ/180=π; gradians=180x400/360=200.
180° = π rad = 200 grad = 0.5 turn.
All values describe the same rotation.
Reproduce this worked caseOpen Angle Converter
Worked case: eighth turn
Convert 45 degrees.
45/360=0.125 turn; radians=45xπ/180=π/4; gradians=50.
45° = π/4 rad = 50 grad = 0.125 turn.
Retaining π/4 preserves an exact radian value.
Reproduce this worked caseOpen Angle Converter
Compare angle-unit conversion cases before generalising
Conversion does not normalize direction or range automatically. An angle of 450° equals 90° modulo a turn but can preserve an extra full rotation in contexts where rotation count matters.
| Degrees | Radians | Gradians | Turns |
|---|---|---|---|
| 180° | π | 200 | 0.5 |
| 45° | π/4 | 50 | 0.125 |
angle-unit conversion: calculation checklist
- Identify source unit
- Use one-turn equivalence
- Retain exact π form
- Distinguish conversion from normalization
- Preserve signed direction
Practical questions
Frequently asked questions
Why is 180 degrees equal to π radians?
A half-circle arc has length πr, and dividing by radius gives π radians.
Are radians a unit?
They are a named dimensionless ratio used to communicate angle clearly.
What is a turn?
It is one complete rotation, equal to 360 degrees or 2π radians.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.