Convert Degrees (°) to Radians (rad)

What Is a Degree?

The degree (°) is a non-SI unit of plane angle defined as exactly 1/360 of a full rotation. It is accepted for use with the SI due to its importance in navigation, surveying, engineering, and astronomy.

The 360-degree circle dates back to ancient Babylonian astronomy (circa 2000 BCE), likely chosen because 360 is highly divisible (by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180). Today the degree remains the dominant angular unit in everyday applications — compasses, protractors, GPS coordinates, and construction.

Key equivalences: 1° = π/180 rad ≈ 0.01745 rad = 60′ (arcminutes) = 3,600″ (arcseconds).

What Is a Radian?

The radian (rad) is the SI derived unit of plane angle. It is defined as the angle subtended at the center of a circle by an arc equal in length to the radius. A full circle equals 2π radians (≈ 6.2832 rad).

The geometric intuition: imagine unwrapping the radius along the circle's edge — the angle that arc covers is exactly 1 radian. Since the full circumference is 2πr, it takes exactly 2π radians to go all the way around. Half a circle (a straight line) is therefore π radians = 180° — the most fundamental relationship between the two units.

Although mathematically dimensionless, the radian is retained as a named SI unit to preserve clarity in physics, calculus, and engineering. Trigonometric functions in mathematics and programming languages natively use radians — sin(π/2) = 1, not sin(90).

Key equivalences: 1 rad ≈ 57.2958°; π rad = 180°; 2π rad = 360°.

Degrees to Radians Formula

The conversion is defined by the relationship π rad = 180°:

radians = degrees × π / 180 exact

This comes from the fact that a full circle is 360° = 2π rad, so 1° = 2π/360 = π/180 rad.

Alternatively, for a direct decimal result without π:

radians = degrees × 0.0174533

Both formulas are equivalent — the first preserves exact fractions of π (preferred in math), the second gives a decimal (preferred in engineering and programming).

Worked Example

Convert 45° to radians:

1. Using π: 45 × π/180 = 45π/180 = π/4 rad
2. Using decimal: 45 × 0.017453 = 0.7854 rad

Both give the same angle. The π/4 form is exact and preferred in mathematics; 0.7854 is what a calculator or program uses internally.

Quick Estimation

For mental math, divide degrees by 57.3 — since 1 rad ≈ 57.2958°. Example: 120° ÷ 57.3 ≈ 2.09 rad (exact: 2π/3 ≈ 2.0944 rad). For small angles under 10°, simply multiply by 0.0175.

Common Conversions at a Glance

Which Is Bigger: Degree or Radian?

A radian is about 57.3 times larger than a degree. One radian spans roughly a sixth of a full circle — the angle where the arc equals the radius. A degree is a tiny 1/360 slice. This is why degree values are always much larger numerically: 90° sounds big, but it is only π/2 ≈ 1.57 radians.

When You Need This Conversion

• Calculus and mathematics: All standard trig derivatives (d/dx sin x = cos x) require radians. A student solving problems with 90° must first convert to π/2 rad before applying calculus formulas.
• Programming: Math.sin(), Math.cos(), and atan2() in JavaScript, Python, C, and most languages expect radians. Converting user input from degrees is a universal first step: rad = deg * Math.PI / 180.
• Physics and engineering: Angular velocity (rad/s), torque equations, and wave functions use radians natively. An engine at 3,000 RPM = 3,000 × 2π/60 = 314.16 rad/s.
• Computer graphics and game development: Rotation matrices, quaternions, and shader functions all use radians. A 45° camera rotation = π/4 rad in the transformation matrix.