Unit Conversions for Fermi Calculation
The unit conversions that appear most often in Fermi estimates — time, distance, area, volume, mass, energy, and speed — with mental shortcuts for each.
Why Units Matter in Fermi Work
A Fermi chain is only valid if the units cancel correctly. The most common errors in estimation aren’t wrong assumptions — they’re silent unit mismatches: mixing km with m, years with seconds, litres with cubic metres. Dimensional analysis is the audit step.
The habit: write units alongside every number, cancel them explicitly, and check that what remains is what you wanted.
Time
| Conversion | Value | Shortcut |
|---|---|---|
| 1 minute | 60 seconds | — |
| 1 hour | 3,600 s | ~4 × 10³ |
| 1 day | 86,400 s | ~10⁵ |
| 1 week | 604,800 s | ~6 × 10⁵ |
| 1 month | ~2.6 × 10⁶ s | ~3 × 10⁶ |
| 1 year | 3.15 × 10⁷ s | π × 10⁷ |
| 1 year | 8,760 hours | ~9 × 10³ |
| 1 year | 365 days | ~4 × 10² |
| Human lifespan | ~75 years | ~2.4 × 10⁹ s |
π × 10⁷ seconds per year is the single most useful time conversion in Fermi work. Accurate to 0.5%.
Working days in a year: ~250 (5 days/week × 50 weeks). Hours worked per year: ~2,000.
Distance
| Conversion | Value |
|---|---|
| 1 km | 1,000 m = 10³ m |
| 1 mile | ~1.6 km |
| 1 foot | ~0.3 m |
| 1 inch | 2.54 cm |
| 1 light-year | ~9.5 × 10¹⁵ m |
| Earth radius | 6.4 × 10⁶ m |
| Earth circumference | 4 × 10⁷ m |
For India-scale estimates: Chennai to Delhi ≈ 2,200 km; India’s land area ≈ 3.3 × 10⁶ km².
Area
| Conversion | Value |
|---|---|
| 1 km² | 10⁶ m² |
| 1 hectare | 10⁴ m² = 0.01 km² |
| 1 acre | ~4,000 m² ≈ 0.4 hectare |
| A city block | ~10⁴ m² (100m × 100m) |
| A football field | ~7,000 m² |
| India land area | 3.3 × 10⁶ km² = 3.3 × 10¹² m² |
When estimating areas: visualise in city blocks (10⁴ m²) or football fields, then scale up.
Volume
| Conversion | Value |
|---|---|
| 1 litre | 10⁻³ m³ = 1,000 cm³ |
| 1 m³ | 1,000 litres |
| 1 cm³ | 1 mL |
| A teaspoon | ~5 mL |
| A cup | ~250 mL |
| A bathtub | ~300 litres |
| An Olympic pool | 2.5 × 10⁶ litres = 2,500 m³ |
| A tanker truck | ~20,000 litres |
The m³ ↔ litre conversion (factor of 1,000) is easy to forget. A cubic metre of water is a tonne, not a litre.
Mass
| Conversion | Value |
|---|---|
| 1 tonne | 1,000 kg = 10³ kg |
| 1 gram | 10⁻³ kg |
| 1 kg of water | 1 litre |
| Human body | ~70 kg |
| A car | ~1,500 kg |
| A loaded truck | ~10,000 kg = 10⁴ kg |
| Earth mass | 6 × 10²⁴ kg |
Water is the universal density anchor: 1 kg/litre = 1,000 kg/m³. Everything denser than water sinks; everything lighter floats. Steel is 8×, concrete is 2.4×, wood is ~0.6×.
Speed
| Conversion | Value |
|---|---|
| 1 m/s | 3.6 km/h |
| 60 km/h | ~17 m/s |
| 100 km/h | ~28 m/s |
| Speed of sound (air) | 340 m/s ≈ 1,200 km/h |
| Speed of light | 3 × 10⁸ m/s |
| Walking speed | ~1.4 m/s ≈ 5 km/h |
| Cycling speed | ~5 m/s ≈ 18 km/h |
Quick conversion: km/h ÷ 3.6 = m/s. Or roughly: km/h × (1/4) ≈ m/s (error ~10%).
Energy and Power
| Conversion | Value |
|---|---|
| 1 kcal (food calorie) | 4,184 J ≈ 4 kJ |
| 1 kWh | 3.6 × 10⁶ J = 3.6 MJ |
| 1 W | 1 J/s |
| 1 kW for 1 hour | 3.6 MJ |
| Daily human food intake | ~2,000 kcal ≈ 8 MJ |
| Human resting power | ~80 W |
Electricity bills are in kWh; physics is in joules. 1 kWh = 3.6 MJ is the bridge.
Dimensional Analysis as a Check
Write units as fractions and cancel:
How many litres of water does India use per day?
Assumption: 150 litres/person/day (urban average)
1.4 × 10⁹ people × 150 L/(person·day)
= 2.1 × 10¹¹ L/day
= 2.1 × 10⁸ m³/day
Units: people × L/(person·day) → the people cancels, leaving L/day. Clean.
If the units don’t cancel to what you want, the formula is wrong — fix it before computing.
Common Conversion Pairs Worth Memorising
These come up in almost every domain:
| From → To | Factor |
|---|---|
| years → seconds | × 3.15 × 10⁷ |
| km² → m² | × 10⁶ |
| m³ → litres | × 10³ |
| km/h → m/s | ÷ 3.6 |
| kcal → joules | × 4,200 |
| kWh → joules | × 3.6 × 10⁶ |
| tonnes → kg | × 10³ |
The goal isn’t to memorise every conversion — it’s to know which ones appear constantly in Fermi chains and have them ready without lookup.