Cycles Per Second to Radians Per Second Converter

How to Convert Cycles Per Second to Radians Per Second

To convert a frequency measurement from cycles per second to radians per second, multiply the frequency value by the conversion factor. Since one cycle per second is equal to 6.283185 radians per second, you can use this formula:

The frequency in radians per second is equal to the cycles per second multiplied by 6.283185.

How Many Radians Per Second Are in a Cycle Per Second?

There are 6.283185 radians per second in one cycle per second.

What Is a Cycle Per Second?

The cycle per second (symbol: cps) is a unit of frequency that measures the number of complete oscillations or cycles occurring in one second. It is numerically identical to the hertz (1 cps = 1 Hz) and is effectively an older name for the same unit. Before the 13th General Conference on Weights and Measures (CGPM) adopted the name “hertz” in 1960, the cycle per second was the standard way to express frequency. The term “cycles per second” was widely used in physics and engineering throughout the first half of the 20th century. The name “cycle per second” has the advantage of being immediately descriptive: it explicitly states what is being counted (cycles) and the time interval (one second). This makes it more intuitive than “hertz” for people unfamiliar with the unit. While the hertz has officially replaced the cycle per second in scientific and technical usage, the term “cycles per second” still appears in older textbooks, patents, and technical manuals. Some engineers and hobbyists continue to use it informally.

One cycle per second is equal to:

• 1 hertz (Hz)
• 0.001 kilohertz (kHz)
• 10⁻⁶ megahertz (MHz)
• 60 revolutions per minute (RPM)
• 6.28319 radians per second (rad/s)
• 360 degrees per second (°/s)

What Is a Radian Per Second?

The radian per second (symbol: rad/s) is the SI derived unit of angular velocity (rotational speed) and also serves as a unit of angular frequency. One radian per second means that the angle swept per second is one radian (approximately 57.296°). In physics and engineering, the radian per second is preferred over revolutions per minute because it simplifies mathematical equations. Angular velocity ω in rad/s is directly related to linear velocity by v = ωr and to centripetal acceleration by a = ω²r. The angular frequency ω (in rad/s) is related to the ordinary frequency f (in hertz) by ω = 2πf. This relationship appears throughout wave physics, signal processing, and electrical engineering. For example, the angular frequency of 50 Hz mains electricity is ω = 2π × 50 ≈ 314.16 rad/s. In control systems engineering, the bandwidth, natural frequency, and gain crossover frequency of systems are all expressed in radians per second. Transfer functions in the Laplace domain use s (= jω) with ω in rad/s.

One radian per second is equal to:

• 0.159155 hertz (Hz)
• 9.5493 revolutions per minute (RPM)
• 57.2958 degrees per second (°/s)
• 1.592 × 10⁻⁴ kilohertz (kHz)
• 0.159155 cycles per second (cps)

Understanding Frequency Units

Frequency is a physical quantity that measures how often a periodic event occurs per unit of time. In the most common usage, it measures the number of complete cycles of oscillation per second. Frequency is the reciprocal of the period (T): f = 1/T.

Different frequency units have evolved to serve different fields and applications. The hertz and its metric multiples dominate physics and electronics, while RPM is standard in mechanical engineering, and radians per second is preferred in mathematical physics and control theory.

Major Unit Families

• SI / Metric units: The hertz (Hz) is the SI unit of frequency, with standard metric prefixes (kHz, MHz, GHz, THz). Each prefix step is a factor of 1,000.
• Cycles per second: The cycle per second (cps) is the original name for the hertz. It is numerically identical (1 cps = 1 Hz) but predates the adoption of the “hertz” name in 1960.
• Revolutions per minute: RPM measures rotational speed and is widely used in mechanical engineering and everyday life. 1 RPM = 1/60 Hz.
• Angular velocity units: Radians per second (rad/s) and degrees per second (°/s) measure angular velocity. One full revolution = 2π rad = 360°, so 1 Hz = 2π rad/s = 360 °/s.

Frequency in Everyday Life

• Sound: Human hearing ranges from about 20 Hz to 20 kHz. Middle C on a piano is 261.6 Hz. Concert pitch A is 440 Hz.
• Electricity: Mains power alternates at 50 Hz (most of the world) or 60 Hz (Americas, parts of Asia).
• Computing: Modern CPUs run at 2–5 GHz. Monitor refresh rates are 60–360 Hz.
• Radio: AM radio: 530–1,700 kHz. FM radio: 87.5–108 MHz. Wi-Fi: 2.4 / 5 / 6 GHz.
• Rotation: Car engines idle at 600–1,000 RPM. Hard drives spin at 5,400–7,200 RPM.

Converting Between Frequency Units

All frequency units measure the same physical quantity (periodic events per unit time), so converting between them involves multiplying by the appropriate conversion factor. The key relationships are:

• 1 Hz = 1 cps (identical units)
• 1 Hz = 60 RPM (one cycle per second = 60 cycles per minute)
• 1 Hz = 2π rad/s ≈ 6.28319 rad/s (one cycle = 2π radians)
• 1 Hz = 360 °/s (one cycle = 360 degrees)

Tips for Frequency Conversions

• For SI metric conversions (Hz, kHz, MHz, GHz, THz), each prefix step is a factor of 1,000. So 1 GHz = 1,000 MHz = 1,000,000 kHz = 10⁹ Hz.
• Hertz and cycles per second are exactly the same unit (1 Hz = 1 cps). The name “hertz” was adopted in 1960 to honour Heinrich Hertz.
• To convert Hz to RPM, multiply by 60. To convert RPM to Hz, divide by 60.
• To convert Hz to rad/s, multiply by 2π (≈ 6.28319). To convert rad/s to Hz, divide by 2π.
• To convert Hz to °/s, multiply by 360. To convert °/s to Hz, divide by 360.
• To convert RPM to rad/s, multiply by 2π/60 (≈ 0.10472). This is one of the most common conversions in mechanical engineering.
• To convert RPM to °/s, multiply by 6 (since 360° ÷ 60 s = 6 °/s per RPM).
• Remember that frequency (f in Hz) and angular frequency (ω in rad/s) are related by ω = 2πf. Many physics equations use ω rather than f.

Cycles Per Second to Radians Per Second Conversion Table

The following table shows conversions from cycles per second to radians per second.

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