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Order-of-magnitude estimates
Intermediate
Fermi estimation gives quick answers to within a factor of ten.
An estimate to within a power of ten, built from rough numbers and simple reasoning (Fermi estimation). The goal is a plausible magnitude, so you can sense-check calculations or see whether an idea is feasible.
How many piano tuners are in a city of 3 million? About one piano per 100 people gives 30,000 pianos, each tuned once a year, a tuner does about 1,000 a year, so roughly 30 tuners.
Don't chase decimals here. The goal is the exponent.
In more detail
An order-of-magnitude estimate gives a result correct to within a factor of about ten, using rounded values and simple reasoning. It is a powerful tool for checking whether a detailed calculation is plausible.
Real-world uses
- Estimating how many piano tuners work in a city, or how much energy a house uses in a year.
- Engineers estimate loads and forces before detailed design to see whether a design is in the right range.
- Before a lab measurement, a student estimates how many water molecules fit in a glass to check that a later calculation is in the right range.
Open the interactive lesson, with simulation and quiz →
Frequently asked questions
How accurate should an estimate be?
Usually within a factor of ten, which is enough to judge whether a result is reasonable.
How do I make a good estimate?
Break the problem into known quantities, use round numbers, and state each assumption clearly.
Why are Fermi estimates useful if they are only rough?
They catch gross errors early. If an answer is off by a factor of a million, the setup is wrong even if the arithmetic is right.
Related topics in Measurement & Math Tools
Deriving formulas by dimensions · Limitations of dimensional analysis · Dimensionless numbers · Unit conversion · Scalars and vectors · Vector representation