The blast radius calculator above evaluates two published relations that underpin explosives safety planning: the Hopkinson-Cranz cube-root scaling law, which reduces charge mass and distance to a single scaled distance, and a peak-overpressure curve expressed as a function of that scaled distance. It exists because standoff distances, magazine separation quantities and evacuation planning are all set from these relations, and because students of structural dynamics meet them early.
Arb Digital builds free physics and engineering calculators that own one job properly. This page is deliberately narrow. It gives a free-field overpressure estimate from a mass and a distance you already have, and it does nothing else. It contains no guidance on obtaining, preparing, handling, placing or using explosives of any kind, and it is not a substitute for the assessment a qualified blast or structural engineer performs.
What This Blast Radius Calculator Does
Blast scaling rests on a simple observation. If you detonate a charge twice as heavy, the volume of air it has to push scales with the mass, and lengths scale with the cube root of the mass. Two explosions of different sizes therefore produce identical peak overpressures at distances in the ratio of the cube roots of their masses. That single idea is the Hopkinson-Cranz law, and it collapses a two-variable problem into one variable.
That variable is the scaled distance, written Z = R ÷ W1/3, where R is the distance and W is the TNT-equivalent charge mass. Everything about the blast wave — peak overpressure, arrival time, positive-phase duration, impulse — can then be plotted against Z alone, which is why published blast parameter charts have scaled distance on the horizontal axis and nothing else.
This calculator returns the scaled distance and the peak incident overpressure that corresponds to it in a widely published closed-form fit, together with the ratio to ambient pressure and the same figure in pounds per square inch, since planning documents on both sides of the Atlantic use different units. The damage description attached to each overpressure band is the standard planning language published alongside these curves, and it is indicative only.
How to Use It
- Enter a charge mass you already have. A licensed net explosive quantity, a published figure from an incident report, or a textbook problem. This page will not help you determine what mass to use for any purpose.
- Set the TNT equivalence factor. Leave it at 1.00 for TNT. If your planning basis is another material, the factor comes from the standard or regulation that governs your site, not from here.
- Enter the distance and choose the burst geometry. A ground burst is conventionally handled by multiplying the charge by about 1.8 to allow for the reflected energy.
- Read the scaled distance first. It is the number that lets you compare your case with any published chart or with the reference calculators used in the field.
- Treat the overpressure as an open-air estimate. It assumes a clear line from source to receiver, with no confinement and no intervening structure.
The Formula: How Scaled Distance and Overpressure Are Calculated
The scaled distance is Z = R ÷ W1/3, in metres per kilogram to the one-third power. The cube root is exactly the operation the cube root calculator performs, and the reason the law is often called cube-root scaling.
The peak incident overpressure is then evaluated from the closed-form Kinney and Graham fit, in which the overpressure ratio is 808 [1 + (Z/4.5)²] divided by the product of √(1 + (Z/0.048)²), √(1 + (Z/0.32)²) and √(1 + (Z/1.35)²), multiplied by the ambient pressure. It is a fit to free-air spherical detonation data and is one of the two relations named in the field as standard, the other being the Kingery and Bulmash polynomial set.
Both relations, and the Hopkinson-Cranz scaling they rest on, are published in International Ammunition Technical Guideline IATG 01.80, Formulae for Ammunition Management, issued by the UN Office for Disarmament Affairs, which is the reference document behind the UN SaferGuard blast parameter calculator used for explosives safety planning. Structural applications of the same parameters, and the codes that govern them, are the subject of the blast-resistant design research programme at the Indian Institute of Technology Bombay department of civil engineering.
The injury thresholds quoted in the damage bands come from the clinical literature on primary blast injury, summarised in the CDC primer Explosions and Blast Injuries: A Primer for Clinicians, which describes the over-pressurisation wave as the mechanism distinguishing high-explosive events from other trauma.
Work a published reference point. At Z = 10 m/kg1/3, the numerator is 808 × (1 + 4.9383) = 4,798.1. The three denominator terms are 208.34, 31.266 and 7.4747, whose product is 48,689. The ratio is 4,798.1 ÷ 48,689 = 0.09854, so the overpressure is 0.09854 × 101.325 = 9.99 kPa, or about 1.4 psi. At Z = 1 the same arithmetic gives a ratio of 9.955 and an overpressure near 1,009 kPa. Both figures sit on the published free-air curve, which is the check this implementation was verified against.
What Free-Field Scaling Cannot Tell You
This is the most important section on the page. Hopkinson-Cranz scaling describes a spherical wave expanding into open air from a point. Almost nothing in the built environment resembles that, and the differences are not small corrections.
Confinement is the largest of them. A wave released inside a room, a corridor, a courtyard or a tunnel cannot expand freely. It reflects off surfaces, the reflections superimpose, and the pressure-time history that results can carry several times the impulse of the free-field case at the same scaled distance, with a long quasi-static phase that free-field curves do not contain at all. Venting, through windows or a lightweight wall, changes that history again.
Reflection matters even outdoors. A wave meeting a rigid surface head-on produces a reflected pressure well above the incident value — a factor of two at low overpressures, and considerably more as the wave strengthens. The incident overpressure this page reports is what a pressure gauge oriented side-on to the flow would read. It is not what the face of a wall experiences, and a facade assessment uses reflected pressure and impulse, not this number.
Geometry, shielding and ground conditions complete the list. Intervening buildings shadow some receivers and focus energy onto others. Soft ground absorbs energy that hard standing reflects. Terrain channels waves along streets. None of this appears in a one-variable scaling law, and all of it is why blast assessment for an actual structure is carried out by a qualified engineer against the applicable code rather than from a chart.
Reading the Damage Bands Honestly
Published planning documents attach descriptions to overpressure ranges, and this tool reports them, but they deserve care. They describe typical outcomes for typical construction, and both of those words are doing a great deal of work. The same overpressure that leaves a modern reinforced frame standing can collapse an unreinforced masonry wall, and glazing behaviour depends on the pane, the frame and the film far more than on a threshold figure.
The injury thresholds behave differently again, because human tolerance depends on the duration of the positive phase as much as on peak pressure. A short, sharp pulse and a long, gentler one at the same peak are not equivalent, which is why injury criteria in the literature are drawn as pressure-impulse curves rather than single numbers. Eardrum and lung injury thresholds quoted as bare pressures are convenient shorthand for a two-dimensional criterion.
The practical consequence is that a damage band is a screening statement, not a prediction. It is useful for deciding that a receiver is clearly outside any zone of concern, or clearly inside one, and it is not useful for arguing that a particular structure will or will not survive. Anything approaching the latter question requires the engineer, the structural drawings and the governing code.
Why the Answer Falls So Steeply With Distance
Overpressure does not fall as a simple inverse-square law. Close to the source the fit behaves roughly as one over Z cubed, out at moderate scaled distances it flattens to something nearer one over Z, and the transition is gradual. That is why the three bracketed terms exist in the denominator of the relation: each one switches on across a different range of Z to reproduce a curve that changes slope.
The cube root in the scaling has a consequence worth stating plainly, because it is counter-intuitive in both directions. Doubling the charge mass moves a given overpressure contour outwards by only 26 per cent, since the cube root of two is 1.26. Multiplying the charge by eight only doubles the distance. Standoff is therefore an extremely effective variable and mass is a comparatively weak one, which is precisely why site security work concentrates on keeping distance rather than on anything else.
Because the relation is a ratio to ambient pressure, altitude also shifts the absolute answer. The air pressure at altitude calculator gives the ambient figure to enter for an elevated site, and the pressure converter and psi to bar converter handle moving the result between the units different standards use.
Where This Sits Next to the Other Pressure Tools
This page owns one narrow calculation: scaled distance and free-field peak incident overpressure. The pressure calculator handles force over area in the ordinary static sense, the hydrostatic pressure calculator handles pressure in a column of fluid, and neither has anything to do with transient waves.
On the wave side, the speed of sound calculator gives the propagation speed of an ordinary acoustic disturbance, which a shock front exceeds — that is what makes it a shock rather than a sound. The sound attenuation calculator and the decibel calculator handle acoustic levels and their logarithmic arithmetic, a different regime entirely from the pressures on this page.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Mixing the two scaled-distance units — feet per pound to the one-third power is about 2.52 times the metric value, so a chart read in the wrong system is wrong by a large factor.
- Using incident pressure where reflected pressure belongs — a surface facing the wave experiences considerably more than the side-on value this page reports.
- Applying free-field curves indoors — confinement changes both the peak and the impulse, and the free-field result can badly understate the loading in an enclosed space.
- Treating a damage band as a prediction — it describes typical construction under typical conditions and cannot speak to a specific building.
- Ignoring duration and impulse — injury and structural response depend on the pressure-time history, not on peak pressure alone, which is why real criteria are drawn as curves.
Related Free Tools From Arb Digital
The cube root calculator performs the scaling operation on its own if you want to see it in isolation. For the pressure units, use the pressure converter and the psi to bar converter; for an ambient figure at an elevated site, the air pressure at altitude calculator. For static and fluid pressure rather than transient waves, use the pressure calculator and the hydrostatic pressure calculator. For ordinary acoustics, use the speed of sound calculator, the sound attenuation calculator and the decibel calculator. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
It is the distance from the source divided by the cube root of the TNT-equivalent charge mass, usually written Z and quoted in metres per kilogram to the one-third power. Hopkinson-Cranz scaling says that two explosions of different sizes produce the same peak overpressure at the same scaled distance, which is what makes published blast charts possible.
Because the energy released fills a volume, and volume scales with the cube of a length. A charge eight times heavier therefore reaches the same overpressure at twice the distance, not at eight times. This is why standoff is such an effective variable and charge mass is a comparatively weak one.
No. This page reports incident, or side-on, overpressure, which is what a gauge aligned with the flow would read. A surface facing the wave head-on experiences reflected pressure, which is at least twice the incident value at low overpressures and considerably more at high ones. Facade assessment uses reflected pressure and impulse.
No. Free-field scaling assumes a wave expanding into open air. Inside a room or a corridor the wave reflects off surfaces, the reflections add, and a long quasi-static phase appears that free-field curves do not contain. Confined-space loading is a separate analysis and can be far more severe at the same scaled distance.
It converts a charge mass expressed for one material into the TNT basis that the scaling curves are built on. This page does not publish factors for any material; if your planning basis is not TNT, the factor should come from the standard, regulation or licence that governs your site.
Because a detonation on a hard surface reflects energy back upwards instead of losing it into a lower hemisphere, so the wave that travels outwards is stronger than free-air scaling would give. Multiplying the charge by about 1.8 is the conventional planning allowance for a perfectly rigid surface, and real ground reflects less.
No. A free-field scaling law contains no information about confinement, reflection, shielding, structural type or connection detailing, all of which dominate real response. Assessment of an actual structure is carried out by a qualified blast or structural engineer against the code that applies in your jurisdiction.
This tool is provided for education, emergency planning familiarisation and structural standoff study only. It evaluates published free-field scaling relations and cannot represent confinement, reflection or site geometry. It is not design guidance. Any assessment of a real site, structure or separation distance must be carried out by a qualified blast or structural engineer, and explosives storage, transport and use are regulated activities governed by your national explosives authority and its licensing regime.
