A gradian (grad or gon) is 1/400 of a circle — so 100 grad is a right angle, 400 grad is a full circle. Introduced during the French Revolutionary metric reform of the 1790s alongside the meter and kilogram. Almost nobody adopted it outside French surveying, mining, and some European CAD software (especially older AutoCAD installations). If you've never seen the GRAD mode on your scientific calculator before, you're not alone — most users go their whole lives without needing it.

How are arc-minutes and arc-seconds used?

In astronomy, surveying, and high-precision navigation. 1° = 60 arc-minutes (60′) = 3,600 arc-seconds (3,600″). The Moon appears roughly 30 arc-minutes wide from Earth; the brightest star Sirius has an apparent diameter of about 6 milli-arc-seconds. A GPS coordinate precise to 1 arc-second nails your position to within 30 meters. Land surveyors in Pakistan use total stations that measure to fractions of an arc-second for property boundary work.

Why does a circle have 360 degrees?

Because ancient Babylonian mathematics used a base-60 number system, and 360 is a highly composite number divisible cleanly by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360 — extremely practical for dividing a circle into equal parts. The approximate match with the 365-day solar year almost certainly helped its adoption. Despite metric reforms, no civilization has managed to dislodge degrees from everyday use.

Are radians used in programming?

Universally. JavaScript's Math.sin(), Python's math.sin(), Java's Math.sin(), and C/C++'s sin() all expect radians. Passing degrees directly gives wrong answers silently — Math.sin(90) returns 0.893 instead of the expected 1.0. Always convert first: radians = degrees × Math.PI / 180. This single bug accounts for an enormous fraction of "my game's rotation is broken" Stack Overflow questions.

What is a 'turn' unit?

1 turn = 1 full revolution = 360° = 2π radians. It's popular in rotation-heavy contexts — RPM, pulley ratios, crankshaft physics, gear design — because the math stays clean without π or 360 clutter. A wheel spinning at 1,200 RPM is doing 20 turns per second. Some mathematicians argue "turn" or τ (tau = 2π) should replace π as the fundamental circle constant; the debate is genuinely contested in academic circles.

How small is 1 arc-second in real terms?

About 1/3,600 of a degree. At Earth's radius (6,371 km), 1 arc-second of latitude equals roughly 30.9 meters on the ground. That's why arc-seconds matter for precision GPS (military-grade is sub-arc-second), land surveying for property boundary disputes, and astronomy for tracking exoplanets. Smartphones with civilian GPS typically achieve 3-5 meter accuracy, equivalent to about 100 milli-arc-seconds.

Is the angle unit the same in a calculator's sin() and a spreadsheet's SIN()?

Both Microsoft Excel and Google Sheets SIN() functions expect radians by default. To use degrees, wrap your input with RADIANS(): =SIN(RADIANS(30)) correctly returns 0.5. This trips up countless students and engineers — typing =SIN(30) gives -0.988 (the sine of 30 radians, which is about 1,719 degrees), not the expected 0.5 from sine of 30 degrees.

Premium Converters

Angle Converter

Unit

Trigonometry-friendly converter for degrees, radians, gradians, turns, arc-minutes, and arc-seconds used in math, GPS, and astronomy.

Unit

Angle Converter

Generated on May 23, 2026

From

Quick Answer

Trigonometry-friendly converter for degrees, radians, gradians, turns, arc-minutes, and arc-seconds used in math, GPS, and astronomy.

Overview — Angle Converter

Unit

An angle converter moves between degrees (familiar everyday unit), radians (the math and physics native unit), gradians (surveying and some European engineering), turns (clean for rotation problems), and minutes/seconds of arc (astronomy and precision navigation). While a full circle is 360 degrees, it is also 2π radians, 400 gradians, and exactly 1 turn. Mixing these up causes silent bugs in code — for example, most programming language trigonometric functions (sin, cos, tan) expect radians, not degrees.

Formula

Radians = degrees × π ÷ 180. Degrees = radians × 180 ÷ π. 1 rad ≈ 57.2958°. 1 turn = 360°.

A right angle (90°) = π/2 radians ≈ 1.571 rad — the corner of every square.

A full circle (360°) = 2π rad ≈ 6.283 rad — one complete revolution of a wheel.

The sun moves 15° across the sky every hour (360° ÷ 24 h) — the basis for sundials.

By Muhammad Salman SaleemUpdated: May 23, 2026MethodologyFree, no sign-upRuns in your browser — your data stays private

From the editors — what to know before you start

Angle conversions usually show up at one of three moments — when you're solving an A-level or B.Eng physics problem and the formula needs radians but your input is in degrees, when you're writing JavaScript or Python code calling Math.sin() and your sprite is rotating to the wrong angle because you forgot to convert from degrees, or when you're reading an astronomy article that quotes positions in arc-seconds and you have no instinct for how small those numbers are. The radian is the mathematician's native unit because it makes calculus and trigonometry clean — the derivative of sin(x) is cos(x) only when x is in radians, full stop. Every programming language's trig functions expect radians; Math.sin(90) returns 0.893, NOT 1.0, because it thinks you mean 90 radians (about 5,156 degrees). Convert first: radians = degrees × π / 180. The degree (360 per circle) won out culturally because the Babylonians used base-60 math and 360 is the friendliest highly composite number for dividing circles into equal parts. Gradians (400 per circle) were a French Revolutionary metric reform that nobody outside surveying ever adopted. Honest take: in everyday Pakistani life, degrees are all you'll ever encounter. The radian becomes urgent the moment you start writing code, solving physics homework, or building anything that rotates.

Pro tips most people learn the hard way

Programming trig rule: every Math.sin(), Math.cos(), Math.tan() call in JavaScript, Python, Java, and C expects RADIANS, not degrees. Always wrap: Math.sin(degrees * Math.PI / 180). This single bug accounts for a huge fraction of "my rotation is wrong" issues in game and animation code.
Microsoft Excel and Google Sheets SIN() functions also expect radians by default. =SIN(30) gives -0.988 (sine of 30 radians), not 0.5 (sine of 30 degrees). Always wrap with RADIANS(): =SIN(RADIANS(30)) for the expected answer. This trips up engineering and finance students constantly.
Memorise the common conversions for physics homework: 30° = π/6 rad, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π. These five values cover 80 percent of textbook problems and skip the calculator entirely.
Arc-seconds matter for GPS and surveying. 1 arc-second of latitude ≈ 30 meters on the ground in Pakistan. Consumer GPS achieves 3-5 meter precision (about 100 milli-arc-seconds). Land disputes in rural Pakistan often hinge on surveyor measurements precise to fractions of an arc-second — taken with total stations that cost more than the disputed plot.
Pro tip for understanding rotation in any context: 1 turn (revolution) is the most intuitive unit. A wheel doing 1,200 RPM is just 20 turns per second. Use "turns" mentally to avoid π and 360 clutter when reasoning about rotation, then convert to degrees or radians only at the final step.

?What is the Angle Converter?

The Formula

Radians = degrees × π ÷ 180. Degrees = radians × 180 ÷ π. 1 rad ≈ 57.2958°. 1 turn = 360°.

A radian is defined as the angle at which the arc length equals the circle's radius. That is why a full circle is exactly 2π radians — its circumference (2πr) divided by r. Radians are the natural unit in calculus: d/dx(sin x) = cos x only when x is in radians. Degrees (360 per circle) originate in Babylonian astronomy, which used base-60 math and mapped nicely to the 360-day year approximation. Gradians (400 per circle) were a French Revolutionary metric reform that never fully caught on except in surveying and some CAD software.

Degrees to Radians — Angle Reference Table

The most-used angle conversions for trigonometry, engineering, and programming.

Degrees (°)	Radians (rad)	π Form	Gradians	Turns
0	0.0000	0	0	0
30	0.5236	π/6	33.33	0.0833
45	0.7854	π/4	50	0.125
60	1.0472	π/3	66.67	0.1667
90	1.5708	π/2	100	0.25
120	2.0944	2π/3	133.33	0.333
180	3.1416	π	200	0.5
270	4.7124	3π/2	300	0.75
360	6.2832	2π	400	1

Put It in Perspective

A right angle (90°) = π/2 radians ≈ 1.571 rad — the corner of every square.

A full circle (360°) = 2π rad ≈ 6.283 rad — one complete revolution of a wheel.

The sun moves 15° across the sky every hour (360° ÷ 24 h) — the basis for sundials.

A ship's compass typically uses 360° divided into 32 'points' of 11.25° each.

Practical Examples

90° equals π/2 radians, or 100 gradians — the right angle used everywhere from construction to robotics.

180° equals π radians (≈ 3.14159) — a straight line and the famous constant in one.

One full turn is 360°, or 2π radians, or 400 gradians — any of which can describe a single rotation.

1 degree equals 60 arc-minutes (60′), and 1 arc-minute equals 60 arc-seconds (60″) — used in astronomy for fine angular measurement.

Earth's rotation is 15° per hour (360° ÷ 24), which equals 0.25° per minute — the basis of timezone width.

A GPS position accurate to 1 arc-second corresponds to about 30 meters on the ground at the equator.

Frequently Asked Questions

Because radians make calculus and trigonometry formulas clean — no factors of π/180 cluttering every step. The derivative of sin(x) is cos(x) only when x is in radians. Arc length formula (s = rθ), angular velocity, rotational kinetic energy — all assume radians. Degrees would force you to write π/180 multipliers everywhere, which is exactly the kind of arithmetic error that loses marks in A-level and university physics exams.

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Premium Converters

Angle Converter

Unit

Trigonometry-friendly converter for degrees, radians, gradians, turns, arc-minutes, and arc-seconds used in math, GPS, and astronomy.

Unit

Angle Converter

Generated on May 23, 2026

From

Quick Answer

Trigonometry-friendly converter for degrees, radians, gradians, turns, arc-minutes, and arc-seconds used in math, GPS, and astronomy.

Overview — Angle Converter

Unit

Formula

Radians = degrees × π ÷ 180. Degrees = radians × 180 ÷ π. 1 rad ≈ 57.2958°. 1 turn = 360°.

A right angle (90°) = π/2 radians ≈ 1.571 rad — the corner of every square.

A full circle (360°) = 2π rad ≈ 6.283 rad — one complete revolution of a wheel.

The sun moves 15° across the sky every hour (360° ÷ 24 h) — the basis for sundials.

By Muhammad Salman SaleemUpdated: May 23, 2026MethodologyFree, no sign-upRuns in your browser — your data stays private

From the editors — what to know before you start

Pro tips most people learn the hard way

Programming trig rule: every Math.sin(), Math.cos(), Math.tan() call in JavaScript, Python, Java, and C expects RADIANS, not degrees. Always wrap: Math.sin(degrees * Math.PI / 180). This single bug accounts for a huge fraction of "my rotation is wrong" issues in game and animation code.
Microsoft Excel and Google Sheets SIN() functions also expect radians by default. =SIN(30) gives -0.988 (sine of 30 radians), not 0.5 (sine of 30 degrees). Always wrap with RADIANS(): =SIN(RADIANS(30)) for the expected answer. This trips up engineering and finance students constantly.
Memorise the common conversions for physics homework: 30° = π/6 rad, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π. These five values cover 80 percent of textbook problems and skip the calculator entirely.
Arc-seconds matter for GPS and surveying. 1 arc-second of latitude ≈ 30 meters on the ground in Pakistan. Consumer GPS achieves 3-5 meter precision (about 100 milli-arc-seconds). Land disputes in rural Pakistan often hinge on surveyor measurements precise to fractions of an arc-second — taken with total stations that cost more than the disputed plot.
Pro tip for understanding rotation in any context: 1 turn (revolution) is the most intuitive unit. A wheel doing 1,200 RPM is just 20 turns per second. Use "turns" mentally to avoid π and 360 clutter when reasoning about rotation, then convert to degrees or radians only at the final step.

?What is the Angle Converter?

The Formula

Radians = degrees × π ÷ 180. Degrees = radians × 180 ÷ π. 1 rad ≈ 57.2958°. 1 turn = 360°.

Degrees to Radians — Angle Reference Table

The most-used angle conversions for trigonometry, engineering, and programming.

Degrees (°)	Radians (rad)	π Form	Gradians	Turns
0	0.0000	0	0	0
30	0.5236	π/6	33.33	0.0833
45	0.7854	π/4	50	0.125
60	1.0472	π/3	66.67	0.1667
90	1.5708	π/2	100	0.25
120	2.0944	2π/3	133.33	0.333
180	3.1416	π	200	0.5
270	4.7124	3π/2	300	0.75
360	6.2832	2π	400	1

Put It in Perspective

A right angle (90°) = π/2 radians ≈ 1.571 rad — the corner of every square.

A full circle (360°) = 2π rad ≈ 6.283 rad — one complete revolution of a wheel.

The sun moves 15° across the sky every hour (360° ÷ 24 h) — the basis for sundials.

A ship's compass typically uses 360° divided into 32 'points' of 11.25° each.

Practical Examples

90° equals π/2 radians, or 100 gradians — the right angle used everywhere from construction to robotics.

180° equals π radians (≈ 3.14159) — a straight line and the famous constant in one.

One full turn is 360°, or 2π radians, or 400 gradians — any of which can describe a single rotation.

1 degree equals 60 arc-minutes (60′), and 1 arc-minute equals 60 arc-seconds (60″) — used in astronomy for fine angular measurement.

Earth's rotation is 15° per hour (360° ÷ 24), which equals 0.25° per minute — the basis of timezone width.

A GPS position accurate to 1 arc-second corresponds to about 30 meters on the ground at the equator.

Frequently Asked Questions

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Angle Converter — Trigonometry-friendly converter for degrees | Premium Converters

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Premium Converters

Angle Converter

Angle Converter

Overview — Angle Converter

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Angle Converter?

The Formula

Degrees to Radians — Angle Reference Table

Put It in Perspective

Practical Examples

Frequently Asked Questions

Popular Conversions

Embed this calculator on your site

Browse other categories

Premium Converters

Angle Converter

Angle Converter

Overview — Angle Converter

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Angle Converter?

The Formula

Degrees to Radians — Angle Reference Table

Put It in Perspective

Practical Examples

Frequently Asked Questions

Popular Conversions

Embed this calculator on your site

Browse other categories

Premium Converters

Premium Converters

Angle Converter

Angle Converter

Overview — Angle Converter

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Angle Converter?

The Formula

Degrees to Radians — Angle Reference Table

Put It in Perspective

Practical Examples

Frequently Asked Questions

Why are radians the 'natural' angle unit?

What is a gradian?

How are arc-minutes and arc-seconds used?

Why does a circle have 360 degrees?

Are radians used in programming?

What is a 'turn' unit?

How small is 1 arc-second in real terms?

Is the angle unit the same in a calculator's sin() and a spreadsheet's SIN()?

Popular Conversions

Embed this calculator on your site

More Unit Tools

Browse other categories

Premium Converters

Angle Converter

Angle Converter

Overview — Angle Converter

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Angle Converter?

The Formula

Degrees to Radians — Angle Reference Table

Put It in Perspective

Practical Examples

Frequently Asked Questions

Why are radians the 'natural' angle unit?

What is a gradian?

How are arc-minutes and arc-seconds used?

Why does a circle have 360 degrees?

Are radians used in programming?

What is a 'turn' unit?

How small is 1 arc-second in real terms?

Is the angle unit the same in a calculator's sin() and a spreadsheet's SIN()?

Popular Conversions

Embed this calculator on your site

More Unit Tools

Browse other categories