Degrees per Hour to Radians per Second Converter

This calculator converts an angular speed given in degrees per hour (°/h) into radians per second (rad/s). Enter the value in degrees per hour, and the result is the equivalent angular speed in radians per second, along with the intermediate value in radians per hour. The conversion uses the standard relationship that 360° equals 2π radians and 1 hour equals 3600 seconds.

Formula

The conversion from degrees per hour to radians per second is a two-step unit conversion:

First, convert degrees to radians. Since 180° equals π radians, multiply the value in degrees by π/180. Then, convert hours to seconds by dividing by 3600 (the number of seconds in one hour). Combined, the formula is:

rad/s = (°/h) × (π / 180) / 3600

Where:

• °/h – angular speed in degrees per hour (dimensionless unit of angle per time).
• π – the mathematical constant, approximately 3.14159265.
• 180 – degrees in half a circle, used to convert degrees to radians.
• 3600 – seconds in one hour.

Worked Example

Using the default value of 360 °/h (which is one full rotation per hour, like the hour hand of a clock):

1. Convert degrees to radians: 360 × (π / 180) = 2π radians (approximately 6.2831853 rad).
2. Convert hours to seconds: divide by 3600 s/h.
3. Result: 2π / 3600 ≈ 0.001745329 rad/s.

The calculator shows 0.001745 rad/s (rounded to 6 decimal places). This matches the formula exactly.

Why Convert Degrees per Hour to Radians per Second?

Degrees per hour is a common unit for very slow angular speeds, such as the rotation of a clock’s hour hand (360°/h), the apparent movement of the sun across the sky (about 15°/h), or the drift of a slowly rotating platform. Radians per second is the standard unit of angular velocity in physics and engineering, especially in the SI system. When you need to plug a rotation rate into equations involving torque, power, or rotational kinetic energy, you almost always need rad/s, not °/h. For example, the power (P) transmitted by a rotating shaft is given by P = torque × angular velocity, where angular velocity must be in rad/s for the result to be in watts.

How to Use the Calculator

Enter any positive number of degrees per hour in the input field. The first output, radians per second, is the main answer. The second output, radians per hour, is an intermediate value that can be useful if you want to convert in two steps. The calculator works for any non-negative value; negative values are not allowed because angular speed is typically expressed as a magnitude, though you could use a negative value to indicate direction if you were working with a signed angular velocity. The default of 360 °/h gives a clear reference point: one full rotation per hour.

Common Mistakes and Misconceptions

• Forgetting the 3600 factor: Some people convert degrees to radians but forget to convert hours to seconds, leaving the result in rad/h instead of rad/s. Always divide by 3600.
• Using 360 instead of 180: The conversion factor from degrees to radians is π/180, not π/360. A full circle is 2π radians, so 360° = 2π rad. The factor π/180 is correct for any number of degrees.
• Confusing angular speed with frequency: Angular speed in rad/s is not the same as frequency in hertz (cycles per second). One full rotation is 2π radians, so 1 revolution per second equals 2π rad/s. If you have a value in revolutions per hour, multiply by 2π and divide by 3600 to get rad/s.
• Rounding too early: If you round π to 3.14, the result will be off by about 0.05%. For precise work, keep more digits of π.

Practical Applications and Context

Degrees per hour is rarely used in physics calculations directly, but it appears in everyday contexts:

• Clock hands: The hour hand of a clock moves at 360°/h (or 0.001745 rad/s). The minute hand moves at 360°/h × 60 = 21,600°/h, which is 0.10472 rad/s.
• Earth’s rotation: The Earth rotates once every 24 hours, so its angular speed is 360°/24h = 15°/h, or about 7.27 × 10�� rad/s.
• Astronomy: The apparent motion of stars across the sky is often quoted in arcseconds per second or degrees per hour. Converting to rad/s helps in telescope tracking calculations.
• Industrial turntables: Some slow-rotating machinery, like a lazy Susan in a kiln, might be specified in degrees per hour.

In all these cases, converting to rad/s allows you to use standard formulas for angular motion, such as ω = θ/t, or to compare with other rotational speeds measured in rad/s.

Limitations and Precision

The conversion is exact, but the result is only as precise as the input. The calculator uses a double-precision floating-point approximation of π, so the output is accurate to at least 12 significant digits, but the display rounds to 6 decimal places. For most practical purposes, 6 decimal places is more than enough. If you need more precision, you can use the formula with a more exact value of π. Also note that the calculator only accepts non-negative values; if you need to represent a negative angular speed (e.g., clockwise rotation), you can enter the absolute value and apply the negative sign manually to the result.

Frequently Asked Questions

How many radians per second is 1 degree per hour?

1 °/h = (1 × π/180) / 3600 ≈ 4.84814 × 10�� rad/s. That’s about 4.85 micro-radians per second. It’s a very tiny value, which makes sense because one degree per hour is extremely slow.

What is the difference between rad/s and revolutions per minute (rpm)?

Rad/s measures the angle swept per second in radians, while rpm measures full rotations per minute. One revolution is 2π radians, and one minute is 60 seconds, so 1 rpm = 2π/60 ≈ 0.10472 rad/s. To convert from °/h to rpm, first convert to rad/s, then multiply by 60/(2π).

Can I use this calculator for negative values?

The calculator disallows negative inputs because angular speed is usually a magnitude. If you need to represent direction, you can compute the absolute value and then assign the negative sign to the result. For example, -360 °/h would be -0.001745 rad/s.

Why do physicists prefer radians per second over degrees per hour?

Radians are the natural unit for angles in mathematics and physics because many formulas, such as the derivative of sin(x) or the relationship between linear and angular velocity (v = rω), are simpler when angles are in radians. Degrees per hour is a human-friendly unit for slow rotations, but it requires extra conversion factors in calculations.

How do I convert degrees per hour to radians per second manually?

Multiply the value in °/h by π, then divide by 180, then divide by 3600. For example, 720 °/h: 720 × π / 180 = 4π rad/h, then 4π / 3600 = 0.00349066 rad/s. You can also combine the factors: multiply by π/648000 (since 180 × 3600 = 648000).

What is the angular speed of the Earth in rad/s?

The Earth rotates once every 23 hours 56 minutes 4 seconds (a sidereal day), which is about 86,164 seconds. That gives an angular speed of 2π/86164 ≈ 7.292 × 10�� rad/s. In degrees per hour, that’s about 15.04 °/h, which is close to the 15 °/h you get using a 24-hour day.

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in deg_h | Degrees per hour | °/h | 360 | 0
out rad_s | Radians per second | rad/s | deg_h * (pi / 180) / 3600 | 6
out rad_h | Radians per hour | rad/h | deg_h * (pi / 180) | 6