A radian is the angle subtended at the center of a circle when the arc length equals the radius. A full circle measures 2π radians, which is approximately 6.2832 radians.

How do you convert radians to degrees?

Multiply the radian value by 180/π. For example, 1 radian × 180/π ≈ 57.2958°, and π radians × 180/π = 180°.

How do you convert degrees to radians?

Multiply the degree value by π/180. For example, 90° × π/180 = π/2 radians, and 360° × π/180 = 2π radians.

Why do mathematicians use radians instead of degrees?

Radians are dimensionless and tie directly to the geometry of the circle, which makes calculus and physics formulas cleaner. In radians, the derivative of sin(x) is cos(x) with no extra constant factor.

How many degrees is one radian?

One radian is approximately 57.2958 degrees, or about 57°18′. This is the angle where the arc cut off by the angle is exactly as long as the radius.

What Is a Radian? (Angle Unit Explained)

Definition

A radian is the natural unit of angle. One radian is the angle subtended at the center of a circle when the arc length equals the radius. Wrap a length of string equal to the radius around the edge of a circle, draw two lines from the ends of that arc to the center, and the angle between them is exactly one radian.

Because the full circumference of a circle is 2π times its radius, a complete turn contains exactly 2π radians ≈ 6.2832 rad. That single fact connects every angle measurement directly to the geometry of the circle, with no arbitrary number like 360 in sight.

One radian works out to about 57.2958°. Unlike the degree, the radian is dimensionless: it is a ratio of two lengths (arc divided by radius), so it carries no units of its own.

Radians and Degrees

Degrees and radians describe the same angles in different scales. The table below lists the landmark angles you will see most often, in both units.

Radians	Degrees
0	0°
π/6	30°
π/4	45°
π/3	60°
π/2	90°
π	180°
2π	360°
1 rad	≈ 57.30°

A useful anchor: π radians = 180°. Once you know that, every other value in the table follows by splitting or scaling. Half of π is a right angle, a third of π is 60°, and a quarter of π is 45°.

Converting

Both conversions come straight from π radians = 180°. To go from degrees to radians, multiply by π over 180. To go the other way, multiply by 180 over π.

Degrees to radians: radians = degrees × π/180
Radians to degrees: degrees = radians × 180/π

For example, 90° × π/180 = π/2 radians, and 3 radians × 180/π ≈ 171.89°. For any value you can drop the numbers straight into the angle converter and read both forms at once.

Why Radians Matter

Radians are not just a tidier alternative to degrees — they are the unit that mathematics and physics are actually built on. In calculus, the derivative of sin(x) is cos(x) only when x is measured in radians; switch to degrees and an awkward factor of π/180 appears in every formula. Choosing radians keeps the trigonometric functions clean.

In physics, angular velocity, angular acceleration, and rotational motion are all expressed in radians per second. The relationship between linear and angular speed (v = rω) only holds when ω is in radians. Radians also define the unit circle, where the angle and the arc length along a circle of radius 1 are the same number — the foundation for how sine and cosine are defined.

Whenever you need to move between this natural unit and the everyday degree, the angle converter handles radians, degrees, gradians, and more in one step, so you can keep your work in whichever unit the problem calls for.