What is DMS format for angles?

DMS (Degrees-Minutes-Seconds) is the compound format used for geographic coordinates and precise angular measurement. Example: 47°26'13" = 47 degrees, 26 arcminutes, 13 arcseconds. To convert DMS to decimal degrees: DD = degrees + minutes/60 + seconds/3600. Example: 47°26'13" = 47 + 26/60 + 13/3600 = 47.4369°.

What are milliradians used for?

Milliradians (mrad) are used extensively in military and firearms applications. The NATO mil (milliradian) is used for range estimation and artillery. At 1000 metres, 1 mrad subtends 1 metre (this is the practical definition used in scope adjustments). 1 mrad = 0.001 radians = 0.05730°. There are 2000π (approximately 6283) true milliradians in a circle.

How many radians are in a full circle?

A full circle contains exactly 2π radians (approximately 6.28318 rad). This is because the circumference of a unit circle is 2π. Other key values: quarter circle = π/2 rad = 90°. Half circle = π rad = 180°. Three-quarter circle = 3π/2 rad = 270°.

Angle Converter — Degrees, Radians, Gradians, DMS & More | LazyTools

📐 Angle Converter

Angle Converter — Degrees, Radians, Gradians & DMS

Q: How do I convert degrees to radians?

To convert degrees to radians, multiply by pi/180 (approximately 0.017453). Example: 90° × (π/180) = π/2 radians ≈ 1.5708 rad. 180° = π radians. 360° = 2π radians. The converter above does this automatically as you type.

Q: How do I convert radians to degrees?

To convert radians to degrees, multiply by 180/pi (approximately 57.2958). Example: π radians × (180/π) = 180°. π/2 radians = 90°. 1 radian = 57.2958°. 2π radians = 360°.

Q: What is a gradian (gon)?

A gradian (also called gon or grade) divides a right angle into 100 units instead of 90 degrees. A full circle = 400 gradians. To convert degrees to gradians, multiply by 400/360 = 10/9. To convert gradians to degrees, multiply by 9/10 = 0.9. Gradians are used in surveying, land navigation, and some military artillery applications.

Q: What are arcminutes and arcseconds?

Arcminutes and arcseconds are subdivisions of degrees used for precise angular measurements. 1 degree = 60 arcminutes (arcmin, '). 1 arcminute = 60 arcseconds (arcsec, "). So 1 degree = 3600 arcseconds. Used in astronomy (telescope pointing), GPS accuracy (1 arcsecond ≈ 30m on Earth's surface), and optics. Example: the Moon's angular diameter is about 31 arcminutes.

Q: What are binary degrees?

Binary degrees (also called binary angular measurement, BAM) divide a full circle into 256 units (2^8). Used in computer graphics, game engines, and embedded systems where integer arithmetic is preferred. 256 binary degrees = 360°. 1 binary degree = 1.40625°. 128 binary degrees = 180° = half circle. 64 binary degrees = 90° = quarter circle.

Convert between 13 angle units simultaneously — type in any field and all others update instantly. Includes DMS (degrees–minutes–seconds) compound input, a live unit circle visualiser, trig values, and common angles quick-fill. Free, no account.

13 units live DMS input Unit circle sin / cos / tan Pi notation

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📐 Angle Converter

Type any value — all 13 units + DMS update instantly

Enter an angle in any unit. The unit circle, trig values, and pi notation update live.

DMS — Degrees ° Minutes ’ Seconds ”

° ’ ”

Standard Angle Units

Degreesdeg / °

Radiansrad

Gradians (gon)grad / gon

Milliradiansmrad

Arcminutesarcmin / ’

Arcsecondsarcsec / ”

Pi Radiansxπ rad

Circle Subdivisions

Turns (rev.)tr / rev

Quadrantsquad (90°)

Sextantssext (60°)

Signs (zodiac)sign (30°)

Compass Pointspt (11.25°)

Binary Degreesbd (360/256)

Unit circle

45°

= π/4 rad

Trigonometric values

sin0.7071

cos0.7071

tan1.0000

cot1.0000

sec1.4142

csc1.4142

Common angles — click to load

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📖 How to Use

How to convert angle units

Enter any value

Type an angle into any of the 13 unit fields. All other units, the DMS compound field, the unit circle, and trig values update instantly.

Use DMS input

For geographic coordinates or precise measurements, enter degrees, minutes, and seconds separately in the DMS compound field at the top. All other units update from DMS too.

Load common angles

Click any of the 12 quick-fill buttons (0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 270°, 360°, 1 rad) to instantly load that angle.

Read the unit circle

The unit circle updates live showing the angle position, a sector arc, and dashed projections for the cos (x-axis) and sin (y-axis) values. The exact pi fraction is shown below.

Check trig values

Sin, cos, tan, cot, sec, and csc for the current angle are shown in the trig panel. Undefined values (tan 90°) are flagged clearly.

Copy any value

Click the 📋 button next to any unit to copy its value. The DMS Copy button copies the full formatted DMS string.

✔ Key Features

The most complete free online angle converter

⚡

13 Units + DMS Live

All 13 angle units and the DMS compound field update simultaneously from any input. No button press, no waiting.

📐

Unit Circle Visualiser

Live SVG unit circle showing angle position, sector arc, and dashed sin/cos projection lines. Updates on every keystroke.

📏

DMS Compound Input

Dedicated degrees-minutes-seconds input with separate fields for D, M, S. Reads and writes simultaneously with all other units.

🧮

Trig Values

Live sin, cos, tan, cot, sec, and csc for the entered angle. Undefined values (tan 90°, sec 90°) are flagged with "undefined".

Pi Fraction Notation

Shows exact pi fractions for common angles: π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π, and multiples. Displayed below the unit circle.

📋

Copy Per Field

Individual copy button on each unit row. The DMS Copy button copies the full formatted string (e.g. 45°0'0") to clipboard.

🔔

12 Quick Angles

One-click load for all standard angles: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 270°, 360°, and 1 radian.

🔓

Free, No Account

100% free with no signup, no limits, and no premium tier. Convert as many angles as you need without creating an account.

📐 Angle Guide

Angle Unit Conversion — Degrees, Radians, Gradians & DMS Complete Guide

Angles can be measured in many different units. Degrees are the most familiar — a full circle is 360°. Radians are the standard mathematical unit where a full circle equals 2π radians. Gradians (also called gon) divide a right angle into 100 parts, so a full circle is 400 grad. Each system has advantages depending on the application.

Conversion formulas

Degrees to radians

rad = deg x (π/180)

180° = π rad = 3.14159...

Radians to degrees

deg = rad x (180/π)

1 rad = 57.2958°

Degrees to gradians

grad = deg x (10/9)

90° = 100 grad

DMS to decimal degrees

DD = D + M/60 + S/3600

47°26'13" = 47.4369°

Decimal degrees to DMS

D = int(DD), M = int(frac x 60), S = frac(M) x 60

47.4369° = 47°26'13"

Degrees to arcminutes

arcmin = deg x 60

1° = 60 arcmin = 3600 arcsec

Standard angle reference table

Degrees	Radians (exact)	Radians (decimal)	Gradians	sin	cos	tan
0°	0	0	0	0	1	0
30°	π/6	0.5236	33.33	0.5	0.8660	0.5774
45°	π/4	0.7854	50	0.7071	0.7071	1
60°	π/3	1.0472	66.67	0.8660	0.5	1.7321
90°	π/2	1.5708	100	1	0	undef.
120°	2π/3	2.0944	133.33	0.8660	-0.5	-1.7321
180°	π	3.14159	200	0	-1	0
270°	3π/2	4.7124	300	-1	0	undef.
360°	2π	6.2832	400	0	1	0

Why radians are used in mathematics and programming

Radians are the natural unit for angles in mathematics. The radian is defined so that an arc of length r on a circle of radius r subtends an angle of 1 radian at the centre. This makes calculus formulas for trigonometric functions simpler: the derivative of sin(x) is cos(x) only when x is in radians. Programming languages and scientific calculators use radians by default for sin(), cos(), tan() functions. To use degrees with these functions, you must convert first: sin(degrees * PI / 180).

Gradians (gon) — the metric angle

Gradians (also called gon or grades) were introduced as part of the metric system to make angle arithmetic easier. The key advantage: a right angle is exactly 100 gradians, so right-angle multiples (slopes, compass bearings) are round numbers. Gradians are used in surveying (particularly in Continental Europe), land navigation, and some military applications. A full circle = 400 grad. Half circle = 200 grad. 1 gradian = 0.9 degrees = π/200 radians.

DMS — Degrees, Minutes, Seconds

DMS is the traditional format for geographic coordinates and astronomical measurements. Just as hours are divided into 60 minutes and 60 seconds, degrees are subdivided similarly. The Moon is 384,400 km away; its angular diameter is approximately 31’6” (31 arcminutes 6 arcseconds). GPS coordinates are often expressed as DMS: 51°30’26”N, 0°7’39”W (the Tower of London). Converting DMS to decimal degrees: DD = D + M/60 + S/3600.

Milliradians in military and firearms

The milliradian (mrad) is used for targeting and range estimation. At 1000 metres, 1 mrad subtends approximately 1 metre — this is the practical mil rule used in sniper and artillery applications. Rifle scopes calibrated in mrad allow precise elevation and windage adjustments. NATO uses a slightly different mil (1/6400 of a circle — not a true milliradian) while the Warsaw Pact used 1/6000. This converter uses the true mathematical milliradian (1/1000 of a radian).

❓ FAQ

Angle converter FAQ

How do I convert degrees to radians?▾

Multiply by pi/180 (approximately 0.017453). Examples: 90 x pi/180 = pi/2 = 1.5708 rad. 180 degrees = pi rad. 360 degrees = 2*pi rad. Type degrees into the converter above and radians update instantly.

How do I convert radians to degrees?▾

Multiply by 180/pi (approximately 57.2958). Examples: pi rad = 180 degrees. pi/2 rad = 90 degrees. 1 radian = 57.2958 degrees. 2*pi rad = 360 degrees.

What is a gradian (gon)?▾

A gradian divides a right angle into 100 parts. Full circle = 400 grad. To convert degrees to gradians, multiply by 10/9. To convert gradians to degrees, multiply by 9/10 (= 0.9). Used in surveying and Continental European land navigation.

What are arcminutes and arcseconds?▾

1 degree = 60 arcminutes ('). 1 arcminute = 60 arcseconds ("). So 1 degree = 3600 arcseconds. Used in astronomy, GPS accuracy, and optics. The Moon's diameter is about 31 arcminutes. 1 arcsecond on Earth's surface is about 30 metres.

What is DMS format?▾

DMS = Degrees-Minutes-Seconds. Example: 47 degrees 26' 13" = 47 + 26/60 + 13/3600 = 47.4369 decimal degrees. Used for GPS coordinates, astronomy, and navigation. The DMS input above converts automatically when you change any field.

How many radians in a full circle?▾

Exactly 2*pi radians (approximately 6.28318). Key values: quarter circle = pi/2 = 1.5708 rad = 90 degrees. Half circle = pi = 3.14159 rad = 180 degrees. Three-quarter = 3*pi/2 = 4.71239 rad = 270 degrees.

What are binary degrees?▾

Binary degrees divide a full circle into 256 units (2^8). 256 binary degrees = 360 degrees. 128 = 180 degrees. 64 = 90 degrees. Used in game engines and embedded systems where integer arithmetic is preferred for angle storage.

Angle converter free no signup▾

The LazyTools Angle Converter is 100% free with no signup. Convert between 13 units and DMS simultaneously, see the live unit circle and trig values, load common angles. No account, no limits.

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