Radians to Kilometers: How to Convert

Radians and kilometers measure different things, so you cannot convert one to the other directly. However, if you know the radius of a circle (or sphere), you can convert an angle in radians to the distance along that circle’s circumference (an arc length) in kilometers. This calculator does exactly that: you enter the angle in radians and the radius in kilometers, and it returns the arc length in kilometers.

Formula

The relationship between an angle in radians and the corresponding arc length is:

s = r × R

• s = arc length (kilometers)
• r = angle in radians (dimensionless)
• R = radius of the circle (kilometers)

This works because a radian is defined as the angle subtended at the center of a circle by an arc whose length equals the radius. So an angle of 1 radian always sweeps an arc equal to one radius, regardless of the circle’s size.

Worked Example

Using the default values from the calculator: angle = 1 radian, radius = 6371 km (Earth’s mean radius).

1. Write the formula: s = r × R
2. Plug in the numbers: s = 1 × 6371
3. Multiply: s = 6371 km

The calculator shows 6371.00 km, which matches.

Why Radians Are Special

Radians are a dimensionless unit because they are defined as a ratio of two lengths (arc length divided by radius). This is why you can multiply a radian value by a length to get another length. Degrees, by contrast, are an arbitrary division of a circle into 360 parts; to use them you must first convert to radians (multiply by π/180) or use a different formula.

This property makes radians the natural unit in mathematics and physics. Formulas in calculus, such as the derivative of sin(x), are only simple when x is in radians. In everyday life, degrees are more common for describing angles, but radians are standard in scientific and engineering contexts.

How to Choose the Radius

The radius you use depends on the situation:

• For a point on Earth’s surface, use the Earth’s mean radius (about 6371 km). If you need more precision, use the local radius (varies from about 6357 km at the poles to 6378 km at the equator).
• For a satellite orbit, use the orbital radius (distance from Earth’s center to the satellite, not the altitude above the surface).
• For any other circle or sphere, use the actual radius of that object.

If you are measuring a distance along a great circle (like a flight path), the arc length formula applies directly. For small angles, the arc length is nearly the same as the straight-line chord distance, but they diverge as the angle grows.

Common Mistakes

• Mixing units: If your radius is in miles, the result will be in miles, not kilometers. Convert the radius to kilometers first.
• Forgetting to convert degrees: If you have an angle in degrees, you must convert to radians (divide by 180 and multiply by π) before using this formula.
• Using the wrong radius: For Earth, using the radius in kilometers (6371) is correct, but some sources give the radius in meters (6,371,000). Double-check your units.
• Confusing arc length with straight-line distance: The formula gives the distance along the curve, not the straight-line (chord) distance between two points.

Limits of the Formula

The formula s = r × R is exact for any angle on a perfect circle. However, it assumes a perfectly spherical Earth. The Earth is an oblate spheroid, so the radius varies with latitude. For most purposes, the mean radius is accurate enough, but for high-precision navigation, you would use a more complex geodetic model.

Also, if the angle is very large (more than 2π radians, i.e., more than one full revolution), the arc length will be longer than the circumference. That is mathematically correct, but in physical terms you might want to take the angle modulo 2π to get the shortest distance along the circle.

Frequently Asked Questions

Can I convert radians to kilometers without a radius?

No. Radians measure angle, and kilometers measure length. Without a radius, there is no way to relate them. The radius is the missing link that turns an angular measure into a linear distance.

How many kilometers is 1 radian on Earth?

On Earth’s surface, 1 radian corresponds to an arc length equal to Earth’s radius, about 6371 kilometers. That is roughly the distance from New York to Buenos Aires, or from London to Cairo.

What is the difference between arc length and chord length?

Arc length is the distance along the curved surface (like a flight path). Chord length is the straight-line distance through the sphere (like a tunnel). For small angles they are nearly the same, but for large angles the chord is shorter. The chord length is 2R sin(r/2), where r is in radians.

Why do we use radians instead of degrees in physics?

Radians simplify many formulas because they are a natural unit based on the geometry of the circle. For example, the derivative of sin(x) is cos(x) only when x is in radians. In degrees, you would have to include a constant factor of π/180. This makes radians the standard in calculus, physics, and engineering.

How do I convert degrees to radians?

Multiply the degree value by π/180. For example, 180° × π/180 = π radians, and 90° × π/180 = π/2 radians. Conversely, to convert radians to degrees, multiply by 180/π.

Is the Earth’s radius exactly 6371 km?

No, it is an average value. The Earth is not a perfect sphere; it is slightly flattened at the poles. The equatorial radius is about 6378 km, and the polar radius is about 6357 km. The mean radius used in many calculations is 6371 km.

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in r | Angle | radians | 1 | 0
in R | Radius | km | 6371 | 0
out s | Arc length | km | r * R | 2