Angle Converter — Degrees, Radians & Gradians

Angle Conversion Guide

What do I need to know about Conversion Factors?

Full circle: 360° = 2π radians = 400 gradians = 1 turn. Conversion: degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π. Degrees to gradians: multiply by 10/9. Gradians to degrees: multiply by 9/10. Key radian values to know: 30° = π/6 rad. 45° = π/4 rad. 60° = π/3 rad. 90° = π/2 rad. 180° = π rad. 270° = 3π/2 rad. 360° = 2π rad. Radians are the natural unit for calculus — all standard differentiation and integration rules for trig functions (d/dx sin(x) = cos(x)) only work when the angle is in radians. Standard reference values for trigonometric functions and geometry are also available in <a href="https://247tables.com" rel="noopener">247 Tables</a>. For other measurement conversions, <a href="https://247quicktools.com" rel="noopener">247 Quick Tools</a> handles length, weight, and temperature.

What's the key thing to understand about Arc Length and Sector Area?

Arc length: s = rθ (where θ is in radians, r is radius). Arc length of a 60° sector with radius 5: θ = π/3 rad. s = 5 × π/3 = 5.24. Sector area: A = ½r²θ (θ in radians). 60° sector, radius 5: A = ½ × 25 × π/3 = 13.09 square units. These formulas are cleaner in radians than degrees — one reason radians are preferred in mathematics and physics. If θ is in degrees: arc length = (θ/360) × 2πr. Sector area = (θ/360) × πr². The radian form is simpler because 2π/360 factors cancel cleanly.

Why Radians Exist?

The radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius. Because of this definition, arc length = radius × angle (in radians) — a beautifully simple relationship. In calculus: the derivative of sin(x) is cos(x) ONLY when x is in radians. In degrees: d/dx sin(x°) = (π/180) cos(x°). In physics and engineering, angles in dynamic equations (angular velocity ω = v/r, angular acceleration α) are always in radians per second. Gradians, also called gons, divide a full circle into 400 units instead of 360 degrees or 2π radians, and are occasionally used in surveying and some European engineering contexts.

What's the key thing to understand about Trigonometric Values?

Key trig values for exact answers: sin(0°) = 0, sin(30°) = 0.5, sin(45°) = √2/2, sin(60°) = √3/2, sin(90°) = 1. cos(0°) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 0.5, cos(90°) = 0. tan(0°) = 0, tan(30°) = 1/√3, tan(45°) = 1, tan(60°) = √3. These are essential for non-calculator paper work in GCSE and A-level maths. The values can be derived from equilateral and right-isosceles triangles — drawing these triangles and applying SOHCAHTOA gives exact values without memorisation.

What's the key thing to understand about Gradians and Other Angle Units?

Beyond degrees and radians, a gradian (also called a "gon") divides a full circle into 400 equal parts instead of 360, so a right angle equals exactly 100 gradians. Gradians were designed to fit the metric system, making calculations like "quarter turn = 100" and "half turn = 200" cleaner than their degree equivalents, and they still appear on some European surveying and artillery instruments. Other specialized units include the "mil" used in military and optics contexts (6,400 mils per circle, approximating 1 mil ≈ 1 meter of lateral spread per 1,000 meters of range) and the "turn" or "revolution," equal to a full 360°/2π radians, used when counting rotations of a wheel, motor shaft, or screw. Converting between any of these and degrees or radians uses the same proportional method: (value ÷ units-per-circle) × 360 for degrees, or × 2π for radians.

What's the key thing to understand about Converting Compass Bearings and Navigation Angles?

Compass bearings measure angles clockwise from true or magnetic north, from 0° to 360°, which differs from the standard mathematical convention of measuring angles counter-clockwise from the positive x-axis (east). To convert a bearing θ_bearing into a standard math angle θ_math, use θ_math = 90° - θ_bearing (adjusting by adding 360° if the result is negative). For example, a bearing of 030° (northeast-ish) converts to a math angle of 60°. This conversion matters whenever navigation, surveying, or GPS data needs to be fed into standard trigonometric formulas for distance and direction, since using a bearing directly in sin() or cos() without converting will place points in the wrong quadrant.

What should I know about Angular Velocity and Radians per Second?

Angle conversion becomes especially important when describing rotational speed, since angular velocity can be expressed in several different units that all describe the same physical rotation. The SI unit is radians per second (rad/s), used throughout physics and engineering formulas because it connects directly to linear velocity via v = ωr and to centripetal acceleration via a = ω²r, without needing any extra conversion factor. Everyday rotational speed, however, is usually quoted in revolutions per minute (rpm), familiar from engine speeds, drill speeds, and record players. Converting between the two uses the same radian relationship as angle conversion generally: one full revolution equals 2π radians, and one minute equals 60 seconds, so ω (rad/s) = rpm × 2π/60 = rpm × π/30. For example, a car engine idling at 800 rpm has an angular velocity of about 800 × π/30 ≈ 83.8 rad/s, and a hard drive spinning at 7,200 rpm has an angular velocity of about 754 rad/s. Because formulas in rotational dynamics are written in terms of radians, converting a stated rpm value into rad/s correctly is an essential first step before applying centripetal force, torque, or rotational kinetic energy equations.

Worked Example and Reference Table: Degrees, Radians, and Gradians

Convert 125° to radians and to gradians. For radians: multiply by π/180 → 125 × π/180 ≈ 2.1817 radians. For gradians (where a full circle is 400 grad instead of 360°): multiply by 10/9 → 125 × 10/9 ≈ 138.89 gradians.

Gradians divide the right angle into a clean 100 units (so 90° = 100 grad exactly), which made them popular in surveying and some European engineering traditions for decimal-friendly angle arithmetic — the trade-off is that gradians don't appear naturally in trigonometric identities and calculus the way radians do, which is why radians remain the standard unit in mathematics and physics despite gradians' tidier right-angle value.

Degrees-Minutes-Seconds (DMS) Notation

Before calculators made decimal degrees convenient, angles -- especially in navigation, surveying, and astronomy -- were expressed in degrees, minutes, and seconds of arc, written as D° M' S". One degree splits into 60 minutes of arc, and one minute splits into 60 seconds of arc, mirroring how hours split into minutes and seconds of time. To convert decimal degrees to DMS, keep the whole-number part as degrees, multiply the remaining fraction by 60 to get minutes, keep the whole-number part of that as minutes, and multiply the new remainder by 60 to get seconds. For example, 34.2575°: the whole part is 34°; 0.2575 × 60 = 15.45, so minutes = 15; 0.45 × 60 = 27, so seconds = 27. The angle is 34° 15' 27". GPS coordinates are still frequently published in DMS form (e.g., latitude 40° 44' 54" N), and converting between DMS and decimal degrees is one of the most common angle-conversion tasks outside pure mathematics.

Converting Angles in Programming and CAD Software

Most programming languages' built-in trigonometric functions (sin, cos, tan) expect radians, not degrees, which is a frequent source of bugs for developers converting from degree-based specifications. JavaScript's Math.sin(), Python's math.sin(), and Java's Math.sin() all require radian input, so a value in degrees must first be multiplied by pi/180. CAD and 3D graphics software is split: some tools default to degrees for human-readable input fields but convert to radians internally for matrix rotations.

A practical rule of thumb: whenever a function name is a plain trig function in a general-purpose programming language, assume radians; whenever it is a user-facing field in design or game-engine software, assume degrees. Checking the documentation once per library avoids a whole class of silent rotation errors.