How this instrument works
A blast wave carries a fixed budget of energy outward from the charge. Because that energy fills an expanding sphere, and a sphere's volume grows with the cube of its radius, the distance at which the overpressure falls to any chosen damage threshold grows only with the cube root of the explosive's energy content. R = k·W^(1/3) is the compact statement of that geometry: the scaling constant k folds in the damage threshold, the ambient air, and the units, while W carries all the information about how much explosive is actually present.
This relationship is the Hopkinson-Cranz scaling law, and it is the reason blast-effects engineering can compress an enormous range of charge sizes — from a firecracker to a bunker-buster — into a single curve. A test fired with a 1 kg charge and one fired with a 1000 kg charge produce overpressure fields that are geometrically identical once distance is expressed as a multiple of W^(1/3); only the physical scale changes. That is what lets published quantity-distance tables cover military ordnance, industrial explosives, and improvised devices with the same formula and different constants.
The constant k is not universal — it is calibrated to one specific effect. A k suited to window breakage is smaller than one suited to eardrum rupture, which is smaller again than one suited to structural collapse or lethality, because each of those thresholds sits at a different overpressure. The formula also assumes a roughly spherical free-air burst with no confinement or ground reflection; a charge detonated inside a room or against a hard surface couples its energy differently, and the same k will underestimate the real radius.
- Enter the explosive charge mass, TNT-equivalent — the device's total energy content expressed in kilograms of TNT, converting from grams or tonnes with the unit menu if needed.
- Set the scaling constant for this damage threshold — larger values of k widen the radius for the same charge and correspond to more severe effects such as structural collapse rather than glass breakage.
- Read the estimated damage radius — the distance from the charge center at which that specific threshold is expected to occur.
- Re-run the calculator with a different charge mass to see how slowly the radius grows: an eightfold increase in mass only doubles it, because the scaling is cube-root, not linear.
Worked example — a 1 kg TNT-equivalent charge
A safety officer needs a quick evacuation figure for a suspected 1 kg TNT-equivalent device — roughly the explosive content of a large pipe bomb — against a scaling constant of k = 4, a typical value for the threshold where windows shatter and unreinforced masonry begins to crack. The formula gives R = 4 × 1^(1/3) = 4 × 1 = 4.0 m: inside that radius the threshold is expected to be met or exceeded, and beyond it the overpressure has fallen off enough that it should not be.
Scale the charge up without touching k and the cube root does the rest of the work: an 8 kg charge, eight times the explosive mass, gives R = 4 × 8^(1/3) = 4 × 2 = 8.0 m — only double the radius, not eightfold. That gap between how fast the mass grows and how slowly the radius follows is the entire point of cube-root scaling, and it is exactly why a bomb squad cannot simply multiply a known safe distance by the ratio of two charge weights.
Questions
Why does blast radius scale with the cube root of mass instead of the mass itself?
Because the blast energy released by the charge spreads outward through an expanding sphere, and a sphere's volume grows with the cube of its radius. Holding the energy needed to reach a given overpressure threshold constant, the radius at which that energy density occurs must grow with the cube root of the total energy — which for a given explosive is proportional to its TNT-equivalent mass. This is the Hopkinson-Cranz scaling law that underlies published quantity-distance tables.
What value should I use for the scaling constant k?
Whatever value matches the specific damage effect you are estimating — window breakage, eardrum rupture, and structural collapse each occur at different overpressures and therefore carry different published constants, typically ranging from a few metres per kg^(1/3) for minor damage up to tens of metres per kg^(1/3) for severe structural thresholds. Explosives-safety standards and quantity-distance tables publish k for specific criteria; this instrument does not choose one for you.
Does doubling the explosive mass double the blast radius?
No, and this is the most common misreading of the formula. Because radius scales with the cube root of mass, doubling the charge multiplies the radius by only 2^(1/3), about 1.26 — a 26 percent increase, not 100 percent. Even an eightfold increase in mass, from 1 kg to 8 kg, only doubles the radius, from 4.0 m to 8.0 m. Extrapolating a known safe distance linearly against charge mass will always undersize the true hazard for smaller charges and oversize it for larger ones.
What does 'TNT-equivalent' mean for the charge mass?
It is the mass of TNT that would release the same blast energy as the explosive actually present, since different explosives release different energy per kilogram — C-4 and ANFO, for instance, are not equivalent to TNT kilogram for kilogram. Published equivalence factors convert a real charge into a TNT-equivalent mass, which is what lets a single k and a single formula apply to any explosive rather than one formula per compound.
Does the cube-root law still apply to very large or nuclear-scale explosions?
The same cube-root form holds from grams of explosive up to megaton-class yields, which is exactly why it remains the standard tool across such a wide range in blast engineering. But the constant k must be re-derived for that regime: nuclear detonations add thermal and radiation effects that chemical explosives do not, and a burst height above ground changes the reflected overpressure. A k calibrated for a TNT anti-personnel charge should never be reused at nuclear scale.
What does a charge mass of zero give as a result?
A damage radius of zero, exactly as the formula predicts: with no explosive energy released, there is no blast wave and no distance at which any overpressure threshold is reached. The calculator accepts a zero charge mass and returns r = 0 without error, which is a useful sanity check that the scaling constant alone is not doing any of the work.