How to Convert Angles Online: Degrees, Radians, Turns and Gradians Explained

You can convert angles online in seconds with the free Angle Converter: enter the angle, choose its unit and the unit you need, and 180 degrees becomes 3.14159 radians, half a turn and 200 gradians at once. The tool works with four units — degrees, radians, turns and gradians — and this guide uses their exact relationships throughout, including the key anchor that one radian equals 57.2958 degrees.

Angles look like the simplest conversion in this series and cause a special kind of trouble: the answers are all small, similar-looking numbers, so a wrong one rarely looks wrong. A student in the wrong calculator mode, a surveyor reading gradians as degrees, a programmer feeding degrees into a function that expects radians — each produces a plausible number that quietly bends every result after it. The four units exist for genuinely different reasons. Degrees serve everyday life and navigation, radians serve mathematics, turns serve rotation and machinery, and gradians serve a decimal dream that surveying never quite gave up. Understanding why each unit divides the circle its own way makes the conversions self-checking: you stop converting numbers and start recognising fractions of a circle.

How Do You Convert Angles Online, Step by Step?

1. Open the converter. Go to the Angle Converter on Asad Webs.
2. Enter the angle exactly as given. Copy it from the drawing, problem, instrument or code, keeping its unit attached. The figure 1.57 is a modest angle in radians and a tiny sliver in degrees.
3. Select the unit you have. Choose degrees, radians, turns or gradians to match the source, not the unit you find most familiar.
4. Select the unit you need. Pick the unit the next step of the job expects: degrees for a protractor or a drawing, radians for a formula or a programming function, turns for rotation counts.
5. Check against the circle anchors. A right angle is 90 degrees, 1.5708 radians, a quarter turn and 100 gradians. A straight line is double each of those. If your result puts a right angle anywhere else, a unit has been swapped.
6. Watch the size pattern. Radians are big units, so radian numbers are smaller than degree numbers for the same angle. Gradians are slightly smaller units than degrees, so gradian numbers run a little higher. Turns are the biggest unit of all here, so most everyday angles are fractions below one.

The Four Units and the Relationships Behind Them

Every angle unit is a way of slicing one full circle. Degrees slice it into 360 parts, a number inherited from ancient astronomy that happens to divide neatly by 2, 3, 4, 5, 6 and many more, which is why it survived. Radians slice the circle by its own geometry: one radian is the angle where the arc length equals the radius, and the full circle holds exactly 2 pi of them, about 6.28318. Turns refuse to slice at all — the whole circle is simply one — which makes rotation counting effortless. Gradians slice the circle into 400, so that a right angle is a round 100, an attempt to make angles as decimal as the metric system.

Unit	Symbol	One full turn equals	Where you meet it
Degree	°	360°	Everyday life, navigation, drawings, protractors
Radian	rad	2 pi rad (about 6.28318)	Mathematics, physics, programming libraries
Turn	turn	1 turn	Rotation, motors, animation, cycling and machinery
Gradian	grad	400 grad	Surveying, some European instruments and maps

The anchor row worth memorising is the half and quarter circle, because most real angles are judged against them. A half turn — a straight line, facing the opposite way — is 180 degrees, 3.14159 radians and 200 gradians. A quarter turn — the right angle, the corner of every page and room — is 90 degrees, 1.5708 radians and 100 gradians. From those two rows every conversion factor can be rebuilt: divide 180 by 3.14159 and you recover 57.2958 degrees per radian; divide 90 by 100 and you recover 0.9 degrees per gradian. A converter you can rebuild from anchors is a converter whose answers you can trust at a glance.

Worked Examples: Three Angles Converted Every Way

Example 1 — the straight angle, 180 degrees. In radians: 180 x (pi / 180) = 3.14159 rad. In turns: 180 / 360 = 0.5 turn. In gradians: 180 / 0.9 = 200 grad. One angle, four costumes — and this is the row to test any tool, calculator or formula against before trusting it with anything harder.

Example 2 — the right angle, 90 degrees. In radians: half of the example above, 1.5708 rad. In turns: 90 / 360 = 0.25 turn. In gradians: 90 / 0.9 = 100 grad, the round number the gradian was invented to produce. A student setting out a square corner, a carpenter checking a frame and a surveyor reading a gon-scale instrument are all handling this same angle in three of its four forms.

Example 3 — one radian in degrees. A programming function returns an angle of 1 radian and a designer needs it in degrees to rotate artwork. Multiply by the anchor factor: 1 x 57.2958 = 57.2958 degrees. In turns that is 57.2958 / 360, about 0.159 turn, and in gradians it is 57.2958 / 0.9, about 63.66 grad. Notice how much bigger a radian is than a degree — nearly sixty of them — which is why radian figures always look deceptively small next to degree figures for the same physical angle.

The chart keeps to degrees because degrees are the unit most readers can picture, and it shows the four compass points of angle intuition: an eighth of a turn at 45 degrees, the right angle at 90, the half turn at 180 and the full circle at 360. Every conversion in this guide is a journey between this picture and one of three other slicings of the same circle. When a converted answer feels wrong, placing it on this line — is it nearer the right angle or the half turn than it should be? — usually exposes the error faster than reworking the arithmetic.

Which Unit Belongs in Which Job?

Situation	Unit to use	Why it fits
School geometry and everyday description	Degrees	Everyone pictures 90 and 180; protractors are marked in degrees
Trigonometry, calculus and physics formulas	Radians	The formulas are built for radians; degrees silently break them
Code and graphics programming	Radians, usually	Most maths libraries expect radians; check the function documentation once
Counting rotation in machinery and animation	Turns	Twelve and a half turns is clearer than 4,500 degrees for a spinning shaft
Surveying with gon-scale instruments	Gradians	The instrument already reads in gradians; converting at the end avoids field errors
Bearings and navigation	Degrees	Compass practice is degree-based, measured clockwise from north

Angles Meet the Rest of the Workshop

Angles rarely travel alone. A slope on a drawing pairs an angle with lengths, and turning a gradient into a height difference is a job for the figures in our length conversion guide together with the Length Converter. Rotation pairs an angle with time: a shaft completing a turn every second is also a frequency problem, which is why the Frequency Converter in the next guide of this series counts revolutions per minute alongside hertz. Even a humble clock is an angle device — its hands sweep 360 degrees in known times — and our Time Converter covers the other half of any such puzzle. Seeing these connections matters because real problems arrive mixed: the exam, the drawing and the datasheet all assume you can move between angle, length and rate without stopping to think about which tool owns which piece.

Five Mistakes That Spoil Angle Conversions

• Using a calculator in the wrong mode. The classic. A sine computed in radian mode for a degree angle (or the reverse) gives a smooth, confident, wrong answer. Check the mode indicator before the first calculation of any session, not after the results look odd.
• Feeding degrees into code that expects radians. Most programming maths functions take radians. Passing 90 where 1.5708 was expected rotates objects by a near-full circle instead of a quarter, and the bug looks like broken graphics rather than a unit error.
• Reading gradians as degrees. The two are only about 11 percent apart, close enough that a 200 grad reading taken as 200 degrees survives a casual glance while bending a survey line by 20 degrees. Confirm the instrument scale before recording anything.
• Treating turns as degrees in rotation counts. A motor specified at 2.5 turns per second is not turning at 2.5 degrees per second; it is sweeping 900 degrees per second. The word turn multiplies everything by 360.
• Forgetting that angles wrap around. 370 degrees is the same direction as 10 degrees, and minus 90 degrees is the same direction as 270. A conversion can be arithmetically right and still need wrapping into the 0 to 360 range before a drawing, bearing or animation will accept it.

A Habit That Makes Angles Self-Checking

Before accepting any converted angle, name the fraction of the circle it represents. Is it about a quarter, about a half, just under a full turn? Then confirm the number in the new unit tells the same story: a quarter circle must come out near 1.57 in radians, near 100 in gradians and near 0.25 in turns. This one habit converts every conversion into two independent judgements that must agree, and disagreement is an immediate, specific warning — far better than discovering the slip three formulas later, when the angle has already been multiplied into a length, a force or a position on a map. The converter gives speed; the fraction gives judgement. Use both, in that order, and angles become the most reliable conversion in this whole series rather than the slipperiest.

Frequently Asked Questions

How many radians are in 180 degrees?

3.14159 radians, the value usually written as pi. Halve it for a right angle: 90 degrees is 1.5708 radians. The whole circle is 2 pi radians, about 6.28318.

How do I convert degrees to radians by hand?

Multiply by pi and divide by 180, or use the shortcut factor 0.0174533. For example, 90 x 0.0174533 = 1.5708 rad. To go back, multiply radians by 57.2958 to get degrees.

What is a gradian?

A gradian, also called a gon, divides a right angle into 100 equal parts, so a full turn is 400 gradians. It was designed to make angle arithmetic decimal, and it survives mainly in surveying and some European instruments.

How many degrees are in one turn?

360 degrees. A turn is the whole circle: 1 turn = 360 degrees = 2 pi radians = 400 gradians. Half a turn is 180 degrees, and a quarter turn is the familiar 90-degree right angle.

Why do mathematics and programming use radians?

Because calculus formulas come out clean in radians: the derivative of sine is cosine only when the angle is measured in radians, and arc length is simply radius times angle. Degrees are a human convenience; radians are the natural unit of the circle itself.

What angle mode should my calculator be in?

Match the mode to the numbers in front of you. If a problem states degrees and the answer comes out wildly small or large, a radian-degree mode mix-up is the first thing to check, before redoing any of the real working.

How do I convert gradians to degrees?

Multiply by 0.9, because 100 gradians fill the 90 degrees of a right angle. So 200 grad = 180 degrees, and 50 grad = 45 degrees. The Angle Converter handles gradians alongside degrees, radians and turns in one step.