Degrees vs. Radians: Conversion, Intuition, and When Each Is Used
October 21, 2026
Angles can be measured in degrees, where a full turn is 360°, or in radians, where a full turn is 2π (about 6.283). The degree is a historical convention, based on the ancient Babylonian base-60 system, while the radian is defined by geometry: one radian is the angle at the center of a circle that subtends an arc equal in length to the radius. This definition makes many formulas in mathematics and physics simpler, which is why calculus and engineering use radians almost exclusively.
To convert, use the fact that 180° = π radians: radians = degrees × π/180, and degrees = radians × 180/π.
Skip the math and get your answer instantly:
Open the Scientific Calculator →Common Conversions
One radian equals 180/π = 57.2958°. So 0.5 radians is 28.65°, and 2 radians is 114.6°. Conversions between degrees and radians for angles you will meet frequently are worth memorizing.
A useful shortcut: 30° is π/6, 45° is π/4, 60° is π/3, and 90° is π/2. Each equals π divided by (180 ÷ the angle in degrees).
| Degrees | Radians (exact) | Radians (decimal) |
|---|---|---|
| 30° | π/6 | 0.5236 |
| 45° | π/4 | 0.7854 |
| 60° | π/3 | 1.0472 |
| 90° | π/2 | 1.5708 |
| 180° | π | 3.1416 |
| 270° | 3π/2 | 4.7124 |
| 360° | 2π | 6.2832 |
Why Radians Are Natural
With radians, the length of an arc is simply s = r × θ and the area of a sector is A = ½ r² θ. A circle of radius 5 with a central angle of 1.2 radians has an arc length of 6 and a sector area of 15. In degrees, the same formulas include extra factors of π/180: s = (θ ÷ 360) × 2πr. The derivative of sin(x) is cos(x) only when x is in radians. Otherwise, a factor of π/180 appears. This simplicity is why radians are standard in calculus, physics, and programming languages.
Angular speed is also in radians per second: a wheel rotating at 60 revolutions per minute turns at 60 × 2π ÷ 60 = 2π rad/s, about 6.28 rad/s.
Where Degrees Remain
Degrees are used in everyday contexts: compasses, maps, construction angles, and weather. Surveyors may use gradians, in which a right angle is 100 grad, and military applications sometimes use mils. Latitude and longitude are given in degrees, minutes, and seconds, where 1 degree = 60 minutes and 1 minute = 60 seconds. To convert 40°26′46″ to decimal degrees: 40 + 26/60 + 46/3600 = 40.4461°.
Avoiding Mode Errors
Scientific calculators and programming languages differ in their defaults. Spreadsheets and most programming languages use radians for trig functions, so SIN(30) returns -0.988 in Excel, not 0.5. To use degrees, convert: SIN(RADIANS(30)) = 0.5. Always check the mode on a physical calculator (DEG, RAD, GRAD) before starting. If your trigonometry results look wrong, the mode is the first thing to check.
A sanity test: sin of 90 should be 1 in degree mode and about 0.894 in radian mode.
Estimating Quickly
One radian is about 57°, a bit under 60°. So π/3 (60°) is just over 1 radian. A full circle has about 6.28 radians, and a right angle is 1.57 radians. For small angles measured in radians, sin θ ≈ θ and tan θ ≈ θ, which is useful in physics and optics. At 5° (0.0873 radians), sin is 0.0872, a difference of only 0.1%.
Frequently Asked Questions
How many degrees is one radian?
About 57.2958°.
How do I convert degrees to radians?
Multiply by π and divide by 180.
Why do mathematicians prefer radians?
They make formulas such as arc length and derivatives of trig functions simple.
How many radians are in a full circle?
2π, about 6.283.
Does Excel use degrees or radians?
Radians. Use RADIANS() to convert from degrees.