To convert radian into degree, change the unit while preserving the signed magnitude. A direction diagram may wrap around a circle, but it should not replace the actual converted number.
The original angle inspector below separates the input, formula, numeric output, and any wrapped direction. The local HTML example pairs worked checks with an interactive unit conversion; it does not record an actual measurement or sensor reading.
Use the full-circle relationship
NIST states the relationship 360° = 2π rad and identifies a right angle as π/2 radians. Dividing the full-circle relationship by two gives 180° = π rad. NIST, SP 330 Section 5.4.8.
The conversion formulas follow:
Degrees = radians × 180/π.
Radians = degrees × π/180.
For 1 radian, multiply by 180/π to get approximately 57.295779513 degrees. For 90 degrees, multiply by π/180: the exact reference is π/2 radians, with a numeric display of approximately 1.570796327 radians.
Keep the input unit and output unit beside both values. The degree symbol here means a plane-angle unit, not a temperature reading. This inspector does not parse degrees/minutes/seconds, apply map coordinates, or calculate solid angles in steradians.
Make a magnitude-preserving worked record
| Supplied input | Conversion | Exact reference where given | Nine-place numeric display |
|---|---|---|---|
| 1 rad | 1 × 180/π | 180/π degrees | Approximately 57.295779513° |
| 90° | 90 × π/180 | π/2 rad | Approximately 1.570796327 rad |
| 450° | 450 × π/180 | 5π/2 rad | Approximately 7.853981634 rad |
| −90° | −90 × π/180 | −π/2 rad | Approximately −1.570796327 rad |
| 0° | 0 × π/180 | 0 rad | 0 rad |
Use the exact references to explain the examples. A finite numeric display of π/2 is still an approximation; it does not become exact because the screen shows nine digits after the decimal point.
The numeric example computes from the supplied input and π, then rounds for display. It should not feed a shortened output back into another conversion as if it were the original angle.
Label the direction diagram separately
For 450 degrees, the magnitude is one full turn plus 90 degrees. The direction sketch shows the endpoint at the same quarter-turn orientation as 90 degrees, while the numeric output preserves the full angle.
Put “wrapped direction” beside the sketch and “original magnitude” beside the actual value. A simple endpoint arrow alone cannot distinguish 90° from 450°. Include a text description of the extra full turn and the signed magnitude so the diagram cannot silently replace the answer.
Choose and label a positive-angle drawing convention, such as counterclockwise from a rightward starting ray, for this original illustration. A negative input should retain its sign in the calculation. If the diagram uses a modulo-360 orientation, describe that as a separate view, not a new angle value.
For −90°, the exact conversion is −π/2 rad. A direction sketch may place its endpoint where 270° would end under that convention, but the inspector must not relabel the supplied magnitude as positive 270°.
These are numeric and illustrative relationships, not evidence that anything physically rotated.
Reject unsupported expressions without rejecting the math
The example accepts a finite decimal number, including a sign, in the selected unit. It does not need an expression parser. Typing pi/2 should therefore produce an instruction to enter a decimal approximation or use the supplied 90-degree preset.
The expression is mathematically meaningful; it is simply outside the chosen numeric-input format. A preset can show the known π/2 reference beside its approximate numeric result without claiming that arbitrary symbolic expressions are parsed exactly.
Blank, malformed, NaN, or infinite entries should clear the result. An explicitly entered zero remains valid. Do not let an empty field become a zero-angle answer.
Clear the old answer whenever input or direction changes. After converting 1 radian, deleting the field should remove 57.295779513°, the formula result, and any prior sketch. Otherwise a viewer may mistake the old output for a conversion of missing input.
Copy this prompt for the complete angle inspector
Create an original self-contained offline HTML degree/radian numeric
inspector in US English, with all necessary local code in one file.
No network calls, storage, sensors or actual rotation/measurement.
Choose conversion direction: radians to degrees or degrees to radians.
Accept finite signed US dot-decimal numeric input only. Zero valid;
blank/malformed/NaN/infinity invalid. No symbolic-expression parser.
Literal pi/2 gets an explanatory error: enter a decimal approximation
or use the supplied 90-degree preset. No guessed numeric prefix.
Formulas: degrees=radians*180/pi; radians=degrees*pi/180.
Compute from source input before rounding. Display numeric outputs
to at most nine fractional places, approximate where rounded, with
nearest rounding and ties away from zero; normalize negative zero.
Preserve signed magnitude, never silently reduce modulo 360.
Include source input/unit, output/unit, formula and plain-text record.
Original checks:
1 rad approximately 57.295779513 degrees.
90 degrees = pi/2 rad, approximately 1.570796327.
450 degrees = 5pi/2 rad, approximately 7.853981634, NOT 90 degrees.
-90 degrees = -pi/2 rad, approximately -1.570796327.
0 degrees = 0 rad; blank is not zero.
Use original direction/arc illustration with a labeled drawing
convention and text alternative. Separate wrapped endpoint orientation
from original angle magnitude. 450 degrees includes one full turn
plus 90; -90 keeps its negative magnitude even if endpoint is shown
at orientation 270. No diagram replaces the numeric conversion.
Clear stale values/formula/sketch on input/direction change or error.
Provide real local Convert/Reset controls, readable labels, keyboard
access, focus/state and narrow layout. If Copy is included, copy
the complete unit-labeled record and show actual success/failure;
do not use a dummy control. Detect unsupported/overflowed results.
No temperature-degree/DMS/map/steradian conversion or persistence.
Return the complete file and manual magnitude/sign/blank/preset
checks; do not claim implementation tests or measurements occurred.
Check the generated inspector with 450° and −90° before relying on its diagram. The numerical record must preserve what you entered, even when a wrapped picture shares an endpoint with a different angle.
Review the angle and its direction in NOS
NOS can save and render prepared HTML. A calculation companion can place the original magnitude, unit conversion, and separately labeled direction sketch together. That helps a reader see why one full turn cannot disappear from a 450-degree answer.
Keep the supplied angle and worked record available, and verify any generated local controls against the examples. Explore NOS.