Circle to Angle Units Converter
Convert an angle between degrees, radians, gradians, sextants, arcminutes, and arcseconds. Enter an angle value, then select the source and target units.
Convert Circle Units to Angle Measures
Circle to Angle Units Formula
Where:
- \(x\) — the angle value entered
- \(f_{\mathrm{from}}\) — degrees per source unit
- \(d\) — the angle expressed in degrees
- \(f_{\mathrm{to}}\) — degrees per target unit
- \(y\) — the converted angle in the target unit
The calculator first expresses the angle in degrees, then converts it to the selected target unit. The degrees-per-unit factors are 1 for a degree, \(180/\pi\) for a radian, 0.9 for a gradian, 60 for a sextant, \(1/60\) for an arcminute, and \(1/3600\) for an arcsecond.
Circle to Angle Units – Calculation Example
Example: use the default settings to convert an angle of 180 degrees to radians.
\[d=180\times1=180^\circ,\qquad y=\frac{180}{180/\pi}=\pi\approx3.141593\text{ rad}\]
Result: 180° equals approximately 3.141593 rad.
How to use the calculator
- Enter the number you want to convert in Angle value. Decimals and negative values are accepted.
- Choose its current unit in From unit: Degree (°), Radian (rad), Gradian (gon), Sextant (1/6 circle), Minute (′), or Second (″).
- Choose the unit you want in To unit. The Converted value shows the result in that unit, rounded to six decimal places.
For example, to convert 2.5 radians to degrees, enter 2.5, select Radian (rad) as From unit, and select Degree (°) as To unit. You can reverse a conversion by swapping the two unit selections.
Understanding circle and angle units
An angle describes an amount of rotation. A complete circle is 360 degrees, \(2\pi\) radians, or 400 gradians. These units describe the same rotation using different scales: a right angle is 90°, \(\pi/2\) rad, or 100 gon. Converting units changes the number shown, not the angle itself.
Radians are common in trigonometry, calculus, and rotational formulas because they relate an angle directly to a circle’s radius and arc length. Degrees are often easier to read in everyday geometry. Gradians divide a right angle into 100 parts and may appear in surveying. A sextant, as defined in this calculator, is one-sixth of a full circle: 60°, or \(\pi/3\) rad. It is an angular unit here, not a measurement from a navigational instrument.
Minute (′) and Second (″) mean arcminute and arcsecond. There are 60 arcminutes in one degree and 60 arcseconds in one arcminute. They measure small angles, not elapsed time. The result is always expressed in the single unit selected under To unit; the calculator does not split an angle into degrees, minutes, and seconds.
Assumptions and practical tips
The converter treats each entry as an angular measurement, including entries larger than one full circle or smaller than zero. For instance, 450° converts to \(5\pi/2\) radians rather than being reduced to 90°. A negative result indicates rotation in the opposite direction under the usual angle convention; it does not change the size of the conversion factor.
Results are rounded to six decimal places for display. If you convert a displayed decimal result back to its original unit, that rounding may produce a small difference. Calculations involving \(\pi\), such as degrees to radians, are generally not exact when written as finite decimals. For manual work, keep an exact form such as \(\pi/2\) rad when possible, and round only at the end.
Angle alone does not tell you how far a point travels or how quickly something rotates: those questions also require a radius or a time interval. For related rotational quantities, see the frequency converter for cycles per unit of time and the torque converter for turning effects of force.
Full-circle reference
| Unit | One complete circle |
|---|---|
| Degrees | 360° |
| Radians | \(2\pi\) rad (about 6.283185 rad) |
| Gradians | 400 gon |
| Sextants | 6 sextants |
| Arcminutes | 21,600′ |
| Arcseconds | 1,296,000″ |
This table provides a quick check on conversions. Half of any listed value represents a half-turn; one-quarter represents a right angle. To convert a fraction of a circle into a unit, multiply that fraction by the unit’s full-circle value.
Frequently Asked Questions
How do I convert arcminutes to radians by hand?
Divide the arcminutes by 60 to get degrees, then multiply by \(\pi/180\). For example, \(30'\) is \(0.5^\circ\), so it equals \(\pi/360\) radians, or about 0.008727 rad. Keeping \(\pi\) in the answer avoids rounding until you need a decimal.
Can I enter an angle written in degrees, minutes, and seconds?
Not as a single entry: this converter expects one number in one unit. First combine the parts into decimal degrees using \(D+M/60+S/3600\). For example, \(12^\circ 30' 36''\) becomes \(12+30/60+36/3600=12.51^\circ\). For a negative angle, apply the minus sign to the entire combined value.
Should I use degrees or radians in a trigonometry calculation?
Use the unit expected by your formula, software, or calculator setting. Most programming-language sine and cosine functions expect radians, while a handheld calculator may have a selectable degree mode. An incorrect mode changes the answer substantially: \(\sin(90^\circ)=1\), but entering 90 into a function expecting radians does not give 1.
Are 0° and 360° the same angle?
They point in the same direction after a full turn, but they represent different amounts of rotation. Angles that differ by a whole number of turns are called coterminal. If you need a direction between 0° and 360°, reduce the angle modulo 360°; if you need the total rotation, retain the full value.
Why does a very small angle convert to 0.000000?
The displayed result is rounded to six decimal places, so a nonzero value smaller than 0.0000005 in the target unit can appear as zero. That does not mean the original angle is zero. For small angles, choose a finer unit, such as arcseconds instead of degrees, or calculate with more precision separately.
How do I turn an angle into a distance along a circle?
You also need the circle’s radius. Arc length is \(s=r\theta\), where \(\theta\) is in radians and \(r\) and \(s\) use the same length unit. For a radius of 2 meters and an angle of \(\pi/2\) radians, the arc length is \(\pi\) meters. This measures distance along the curve, not straight-line distance.
Is a gradian the same as a percent grade?
No. A gradian (gon) measures an angle; percent grade measures slope as vertical change divided by horizontal distance, multiplied by 100. There is no fixed multiplication factor between them. To convert an angle \(\theta\) to percent grade, use \(100\tan(\theta)\), with your tangent function set to the correct angle unit.