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Quick guide
Understand the calculation
Convert angles between 9 units —decimal degree, degrees-minutes-seconds (DMS), degrees and decimal minutes (DDM), arcminute and arcsecond, radian, milliradian, gon and turn— and one inclination between degrees, radians, percent, per mille and a 1 : n ratio. It accepts coordinates pasted as they are, with symbols or a hemisphere letter, and radians written as multiples of π.
When to use this tool
Use it to move a trigonometry exercise from degrees to radians and see its exact form in π; to turn a latitude or longitude in DMS into decimal degrees —or back— for a GIS, a spreadsheet or a GPS; to work with arcseconds in astronomy and optics or with milliradians in alignment and ballistics; and to translate the grade of a road, a roof, a ramp or a topographic profile between percent, degrees and ratio.
How the conversion works
In “Angle”, the value is first expressed in degrees and from there in the unit of the result. The page shows the general formula for the chosen pair and, below it, the same operation with your number. In DMS and DDM you can paste the angle as it appears on a map or in a book: 40° 26′ 46″ N, 40 26 46, 40:26:46 or 40º26'46'', with a decimal point or comma in the last component. A DMS result is rounded before it is split into degrees, minutes and seconds, so it never shows 60″ or 60′: 10.1° is 10° 6′ 0″. In radians you can type 5π/4 or 5pi/4, and when the angle is an exact multiple of π —down to π/360— the result says so: 30° = π/6 rad. If the angle is negative or exceeds one turn, its coterminal between 0 and 360° is added. Between 0° and 360°, and without a hemisphere letter, the angle type is also shown, based on the value as typed rather than the rounded one: zero (0°), acute (under 90°), right (90°), obtuse (between 90° and 180°), straight (180°), reflex (between 180° and 360°) or full (360°). In “Slope”, percent and ratio come from the tangent of the angle: % = 100 × tan θ and n = 1 / tan θ. “Show all equivalents” lists the value in every unit of the mode, and the significant-figures selector switches between 6 and 10.
The tool converts one angle at a time: a coordinate pair is rejected with its own notice, and there is no UTM, datum, quadrant bearing (N 30° E), hour angle or military mil, which has three definitions in use. Minutes and seconds go from 0 to less than 60, only the last component can have decimals and scientific notation is not accepted in DMS or DDM, because there E means east. In “Slope” the inclination is entered as a magnitude below 90°: an 8 % downhill grade is written 8. A reading that cannot be expressed in its format —a DMS above one million degrees or below a thousandth of a second— shows as “—”, and the others remain available.
Verified examples
- Latitude · 40° 26′ 46″ N
40° 26′ 46″ N = 40.4461° - West longitude · 3° 42′ 12.5″ W
3° 42′ 12.5″ W = −3.70347° - Radians with π · 5π/4 rad
5π/4 rad = 225° - Carrying minutes · 10.1°
10.1° = 10° 6′ 0″ - Ramp · 1 : 12
1 : 12 = 8.33333 % = 4.76364°
Frequently asked questions
Why is 10.1° not 10° 5′ 60″?
Because 60″ is a whole minute and has to be carried. That result appears when degrees, minutes and seconds are separated before rounding: 0.1° is exactly 6′, but in floating point 10.1 is 10.0999999999999996…, the seconds come out as 59.9999… and, rounded to two decimals, become 60.00. If the total seconds are rounded first and only then separated, the result is 10° 6′ 0″.
How do I write a southern latitude or a western longitude?
With the DMS or DDM unit, in two equivalent ways: a minus sign in front (-3 42 12.5) or the hemisphere letter at the start or end (3 42 12.5 W, W 3 42 12.5). S and W give negative values, like the decimal degrees of a GIS; do not use a sign and a letter together. With N or S the value cannot exceed 90° and with E or W it cannot exceed 180°. The sign applies to the whole angle: -0° 30′ is -0.5°, not +0.5°.
How many metres does a second or a decimal of a degree represent?
Along a meridian, 1′ of latitude is one nautical mile, about 1852 m, so 1″ is about 31 m. In decimal degrees, the fifth decimal (10⁻⁵°) is about 1.11 m at the equator and the sixth about 11 cm. In longitude those distances shrink with the cosine of the latitude. That is why five decimals already exceed the precision of an everyday GPS, which is a few metres.
Is a 100 % slope vertical?
No. 100 % means you rise as much as you advance: a 1 : 1 ratio, that is 45°. A vertical wall would have an infinite slope: that is why 90° has no percentage or 1 : n ratio. Near vertical, the percentage grows much faster than the angle: 60° is already 173.205 %.