Fermi Estimation

"Can the U.S. power its grid entirely with rooftop solar?" You could spend weeks modeling this, or you could get within a factor of two in 60 seconds with back-of-the-envelope arithmetic. That second approach is a Fermi estimation: breaking an impossible-sounding question into smaller, estimable pieces.

A Fermi estimation decomposes a question into factors you can estimate independently, then multiplies them together. Errors in individual factors tend to cancel (some too high, some too low), so the final answer is usually within an order of magnitude of the true value. The method is named after physicist Enrico Fermi, who famously estimated the energy of the first nuclear test by dropping torn paper scraps and measuring their displacement.

The rules: round aggressively (use powers of 10), state each assumption explicitly, and check whether the final answer passes a sanity test against your scale anchors. If the answer requires more energy than the sun produces, you made an error somewhere.

Worked Example

Can U.S. rooftop solar meet total U.S. electricity demand?

• U.S. electricity demand. About 4,000 TWh/year (scale anchor).
• Available rooftop area. Roughly 130 million buildings, average usable roof ~50 m². That is about 6.5 x 10⁹ m².
• Solar output per m². Average U.S. insolation ~170 W/m². At 20% panel efficiency, that is 34 W/m². Over a year: 34 x 8,760 hours = ~300 kWh/m²/year.
• Total rooftop output. 6.5 x 10⁹ x 300 = ~2,000 TWh/year.

2,000 TWh vs. 4,000 TWh demand. What does this tell you?

Rooftop solar could supply roughly half of U.S. electricity, not all. And this is before accounting for storage, cloudy days, or north-facing roofs. The estimate is rough, but the conclusion is clear: rooftop solar alone cannot do the whole job. You learned this in 60 seconds.

Enrico Fermi didn't need a supercomputer to estimate an atomic blast from scraps of paper. Most energy claims worth checking don't need one either.

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