---
title: "Why Blast Radius Scales With the Cube Root of Yield"
description: "Blast radius scales with the cube root of yield because blast energy fills a volume — 1,000× the yield reaches only 10× further. Hopkinson–Cranz."
url: https://www.shreyanshkumarsingh.com/notes/blast-radius-cube-root-scaling
author: Shreyansh Kumar Singh
updated: 2026-10-08
---

# Why Blast Radius Scales With the Cube Root of Yield

By Shreyansh Kumar Singh · 2026-10-08

**Why does blast damage radius scale with the cube root of yield?**

Because a blast's energy spreads through a volume, and volume grows with the cube of distance: to get the same overpressure at a larger distance R, you need energy proportional to R³, so R grows as the cube root of the energy. That's Hopkinson–Cranz scaling — a thousand times the yield moves any given overpressure contour only about ten times further out.

## The law in one line

Overpressure at distance R from an explosion of yield W depends only on the scaled distance Z = R / W^(1/3). Two explosions produce the same overpressure wherever they have the same Z. Double the radius and you need eight times the yield to reach the same overpressure there.

## What that looks like with real numbers

AEON, the weapons-consequence simulator in Shreyansh's The Evolution, builds its air-blast model on this law from the 1 kt reference curve in Glasstone & Dolan, and its test suite checks it against published figures: the cube-root law to 1e-9, the 5 psi radius of a 1 Mt burst at 7.00 km, and the optimum burst height at 178 m for 1 kt versus 1,780 m for 1 Mt — a thousandfold yield, a tenfold height. One exponent is why a single slider can span seven orders of magnitude of yield.

## Not every effect follows it

Cube-root scaling is a blast law. Other effects scale differently, which is why a consequence model can't use one radius for everything: thermal fluence falls off with the square of distance plus atmospheric attenuation, prompt radiation follows its own exponential attenuation, and crater radius in AEON scales closer to Y^(1/3.4). Each zone in AEON comes from its own published model for exactly that reason.

## Scope

AEON models consequences only — blast, heat, radiation, cratering and fallout — for the same reason public references like The Effects of Nuclear Weapons exist: the numbers are otherwise abstract. It contains no weapon design or targeting information, and its casualty figures are labelled as upper-bound estimates with wide error bars.


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Shreyansh Kumar Singh — AI & Full-Stack Engineer, Pune, India. Contact: shreyanshkumarsingh208@gmail.com or https://www.shreyanshkumarsingh.com/contact (call and WhatsApp buttons). More: https://www.shreyanshkumarsingh.com/llms.txt · https://www.shreyanshkumarsingh.com/sitemap.xml