Convert between degrees and radians for angle measurements. Essential for trigonometry, geometry, and engineering calculations.
Last updated: March 2026 | By Patchworkr Team
Degrees and radians are two units for measuring angles. A degree is 1/360 of a full rotation, while a radian is the angle formed when the arc length equals the radius of a circle.
Radians are the standard unit in mathematics and physics because they simplify many formulas. For example, the derivative of sin(x) is cos(x) only when x is in radians. Most programming languages and calculators use radians for trigonometric functions.
The conversion factor is based on the fact that a full circle is 360 degrees or 2π radians. Therefore, 1 radian = 180/π degrees ≈ 57.2958°, and 1 degree = π/180 radians ≈ 0.0174533 rad.
Convert 90 degrees to radians:
Radians simplify many mathematical formulas, especially in calculus. The derivatives and integrals of trigonometric functions work elegantly with radians, and they naturally connect arc length to radius. In physics and engineering, radians eliminate conversion constants.
A radian is the angle formed when the arc length equals the radius of a circle. Since the circumference of a circle is 2πr, there are 2π radians in a full circle (360 degrees). This makes radians a "natural" unit based on circle geometry.
A full circle contains 2π radians, which equals approximately 6.28319 radians or exactly 360 degrees. A semicircle is π radians (180°), and a quarter circle is π/2 radians (90°).
Degrees are commonly used in navigation, geography, construction, and everyday contexts where angles are more intuitive in 360-degree format. Use radians for mathematical and scientific calculations, especially in calculus, physics, and programming.
Gradians (or grads) divide a circle into 400 units. They were designed for easier decimal calculations: 100 gradians = 90 degrees = π/2 radians. They are rarely used in modern practice but appear in some surveying and European calculators.
A turn (or revolution) is one complete rotation of 360 degrees or 2π radians. It is sometimes used in computer graphics and animation to represent full rotations. Fractional turns (0.25 = quarter turn) are intuitive for rotational motion.
Most scientific calculators have three modes: DEG (degrees), RAD (radians), and GRAD (gradians). Always check which mode is active before trigonometric calculations. Programming languages like JavaScript, Python, and C use radians by default for all trig functions.
Yes! Negative angles represent clockwise rotation, while positive angles are counterclockwise. Angles greater than 360° (or 2π rad) represent multiple rotations. For example, 450° = 360° + 90°, which is one full rotation plus a quarter turn.
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