Last updated: October 2026
The formula: f = c ÷ λ, fully worked
The entire calculator is one equation: v = λ × f, rearranged to f = v ÷ λ or λ = v ÷ f. Here it is with real numbers for red light at 650 nm:
1. Convert the wavelength to meters: 650 nm = 650 × 10−9 m = 0.00000065 m.
2. Divide the speed of light by it: 299,792,458 ÷ 0.00000065 = 461,219,166,154 Hz.
3. That is 461.2 THz (terahertz — trillions of cycles per second).
The reverse is the same idea flipped. FM radio at 100 MHz: λ = 299,792,458 ÷ 100,000,000 = 2.998 m — roughly 3 meters, which is why FM antennas are about a meter and a half long (half a wavelength). The calculator above prints this working under every result, so you can check it or copy it into homework.
Wavelength, frequency, and photon energy
The wave equation tells you the geometry; E = hf (Planck's relation) tells you the quantum side — how much energy each photon packs, where h = 6.62607015 × 10−34 J·s. Combine them and you get E = hc ÷ λ, the most useful form when you start from a wavelength. Worked for a 532 nm green laser pointer:
1. f = 299,792,458 ÷ (532 × 10−9) ≈ 5.635 × 1014 Hz.
2. E = 6.62607015 × 10−34 × 5.635 × 1014 ≈ 3.734 × 10−19 joules.
3. Divide by the electron charge (1.602176634 × 10−19) → ≈ 2.33 eV.
For perspective: a 2.4 GHz Wi-Fi photon carries only about 10 micro-eV — two hundred thousand times less. Around 10 eV is roughly where photons start being able to knock electrons off atoms (ionizing radiation), which is why UV gets its own safety warnings and radio waves don't.
EM spectrum bands: where your result lands
| Band | Wavelength | Frequency | Everyday examples |
|---|---|---|---|
| Gamma rays | < 0.01 nm | > 30 EHz | Nuclear decay, PET scans |
| X-rays | 0.01 – 10 nm | 30 PHz – 30 EHz | Medical imaging, airport scanners |
| Ultraviolet | 10 – 400 nm | 750 THz – 30 PHz | Sunburn, sterilization lamps |
| Visible light | 400 – 700 nm | 430 – 750 THz | Human vision, LEDs, lasers |
| Infrared | 700 nm – 1 mm | 300 GHz – 430 THz | Heat, TV remotes, night vision |
| Microwaves | 1 mm – 1 m | 300 MHz – 300 GHz | Wi-Fi, 5G, radar, ovens |
| Radio waves | > 1 m | < 300 MHz | FM/AM radio, long-wave |
Boundaries are approximate — physicists argue about the exact edges (the radio/microwave line especially). For visible light the calculator goes further and names the color: 650 nm is red, 532 nm is green, 450 nm is blue.
Light vs. sound: same equation, different speed
Nothing about the math changes for sound; only the speed does, and unlike light, the speed of sound is not constant. The standard approximation is v ≈ 331.3 + 0.606 × T (T in °C): about 331 m/s at freezing, 343 m/s at room temperature, 349 m/s on a hot 30 °C day. That is a ~5% swing, enough to matter in acoustics and enough to get a homework answer marked wrong. Select “Sound in air” in the calculator and enter the temperature — it adjusts automatically. (Underwater, sound travels at roughly 1,500 m/s: switch to “Custom speed” and enter 1500.)
Wavelength and frequency, properly understood
Think of waves passing a fixed point, like fence posts seen from a moving train. Wavelength (λ) is the distance between two adjacent posts. Frequency (f) is how many posts flash past each second. Speed (v) is how fast you are moving. If the posts are fixed in the ground (constant v) and they start flashing past twice as fast, the posts must be half as far apart — that is the inverse relationship, v = λf, with no deeper mystery.
One subtlety worth knowing: frequency is the wave's identity; wavelength is its outfit. When light enters glass, its wavelength shrinks but its frequency does not change — the color you see is set by frequency, which is why “650 nm red” stays red in any medium. Radio engineers therefore talk in MHz (identity), while optics people talk in nm (outfit); they are describing the same wave.
Units cheat-sheet (the mistakes live here): 1 km = 103 m, 1 m = 1 m, 1 cm = 10−2 m, 1 mm = 10−3 m, 1 μm = 10−6 m, 1 nm = 10−9 m. And 1 THz = 1012 Hz, 1 GHz = 109 Hz, 1 MHz = 106 Hz, 1 kHz = 103 Hz. The classic error is dividing the speed of light by nanometers directly — that gives an answer a billion times too large. Always convert to meters first; this calculator does it for you.
Period, the last piece of vocabulary: T = 1/f is the time one cycle takes. A 100 MHz FM wave has a period of 10 nanoseconds; middle C's period is about 3.8 milliseconds. The calculator shows it alongside every result.