Kidney Stone Volume and Burden Calculator

Stone burden from three CT diameters, by all three published ellipsoid formulas at once. A stone is reported by its longest diameter and behaves like a volume, so a 10 mm stone is eight times the burden of a 5 mm one rather than twice.

Last updated 8 Oct 2026

Kidney Stone Volume and Burden

Three diameters → volume

Longest diameter (mm)The three fields take three orthogonal diameters, in any order: the scalene formula in the headline is a symmetric product, so a transposed pair cannot change the answer. Measure all three on the same series — CT and plain-film measurements of the same stone are not the same numbers, and a stone measured on a coronal reformat is not the stone measured on an axial slice.

Second diameter (mm)

Third diameter (mm)Three radiologists measuring the same stone linearly differed by an average of 26.3 per cent in one series, and by 1.2 to 1.9 mm in another. That uncertainty is cubed by this formula, so treat the volume as approximate by tens of per cent.

183mm³Example

A CT reporting a stone of 10 × 7 × 5 mm

Three published formulas, and the cube that matters

Scalene (the headline) V = π/6 × a × b × c
Oblate V = π/6 × a × a × c
Prolate V = π/6 × a × b × b
Equivalent sphere diameter = ∛(a × b × c), which is the geometric mean of the three
π/6 = 0.5236, so a sphere of diameter d has volume 0.5236 d³

the cube relationship: Burden scales as a volume and not as a diameter, so doubling every dimension multiplies the burden by eight rather than by two. A 5 mm stone is 65 mm³; a 10 mm stone is 524 mm³; a 15 mm stone is 1,767 mm³ — eight times and twenty-seven times, not two and three. This is the single most useful thing on the page, and it is why ‘the stone has only grown 2 mm’ can describe a doubling of burden
which variant to use: the series that compared them found the oblate and scalene forms correlated with software-segmented volume at r above 0.9 and the prolate form above 0.8, in 72 adults; the authors reported that the three gave significantly different volumes for the same stone. All three are printed here so the choice is visible. The oblate and prolate forms use one of the three diameters twice, so unlike the scalene form they are NOT symmetric — which of your measurements you put in which field changes their answers, and that is a property of those published formulas rather than of this page
the equivalent sphere diameter: the diameter of the sphere with the same volume, which is exactly the cube root of the product of the three diameters. It is the honest single number to quote when a guideline’s strata are linear, and it is always between the smallest and the largest of the three measurements
the formulas overestimate, increasingly with size: against physically determined volumes, best-fit ellipsoid formulas overestimated by 27 per cent for stones under 9 mm and by 89 per cent for stones of 20 mm or more. One published comparison against a 3D segmentation showed a 41 per cent ellipsoid overestimate. So this number is a burden INDEX for comparison, not a physical volume
the stone-risk panel is elsewhere: The 24-hour urine stone-risk panel and its interpretation already live in the urine chemistry category and are not rebuilt here. The 24-hour urine stone-risk interpreter reads a collection against the EAU’s own limits, and the urine collection adequacy calculator, urine calcium converter, urine citrate converter, urine oxalate converter, urine uric acid converter and urine creatinine converter are the panel underneath it. A score, an index, a measured volume or an attenuation value is not a diagnosis, and a proportion measured in a cohort is not a probability for one patient.

Worked example

A CT reporting a stone of 10 × 7 × 5 mm
10 × 7 × 5 = 350 mm³ of bounding box
350 × π/6 = 350 × 0.5236 = 183 mm³ by the scalene formula
The oblate variant (10 × 10 × 5) gives 262 mm³ and the prolate variant (10 × 7 × 7) gives 257 mm³ for the same stone — a 43 per cent spread across three published formulas, all shown above
Equivalent sphere diameter = ∛350 = 7.0 mm, which places this stone in the EAU's 5–10 mm stratum rather than in the 10–20 mm one its longest diameter alone suggests
Now double every dimension, to 20 × 14 × 10 mm: the volume becomes 1,466 mm³, exactly eight times larger. The longest diameter doubled; the burden octupled

What doubling a diameter does to the burden

Spheres, for clarity: a real stone is an ellipsoid and its volume is the product of its three diameters times π/6. The column on the right is the reason the EAU’s observation that volume predicts stone-free status better than diameter is not a technicality. A fragment described as ‘4 mm residual’ carries a sixteenth of the burden of a 10 mm stone, and the same series that reports this notes a 16 per cent reintervention rate for residual fragments of 4 mm or less at a median seven months — so a small burden is not no burden.
Sphere diameter	Volume (mm³)	Relative to the 5 mm stone
4 mm	34	0.5 ×
5 mm	65	1 ×
7 mm	180	2.7 ×
10 mm	524	8 ×
15 mm	1,767	27 ×
20 mm	4,189	64 ×

How the published measurements actually behave

Two things sit awkwardly together here and both are true. The volume is the better predictor, and the volume computed from three diameters is substantially wrong — overestimating by the order of 27 per cent in small stones and 89 per cent in large ones, on top of a linear measurement that varies by a quarter between observers. Which is why this number is useful for comparing one stone with another measured the same way, and not for quoting as a physical volume.
Finding	Figure	Source
The EAU’s size strata are LINEAR, by cumulative diameter	Under 5 mm · 5–10 mm · 10–20 mm · over 20 mm	EAU urolithiasis guideline, 2026
Volume predicts outcome better than diameter	The EAU states volume is a better predictor of stone-free status, and that the lack of cut-off values guiding treatment prevents its clinical adoption	EAU urolithiasis guideline, 2026
How much of the volume a maximum diameter captures	About 76 per cent for stones under 10 mm, about 10 per cent for stones over 20 mm	AUANews, May 2024
Ellipsoid overestimate against physical volume	27 per cent under 9 mm, 89 per cent at 20 mm and above	AUANews, May 2024, citing Finch and colleagues
Interobserver error in the linear measurement itself	26.3 per cent between three radiologists; 1.2 to 1.9 mm in a second series	AUANews, May 2024, citing Patel and Eisner
Explanatory value for shockwave lithotripsy outcome	Nagelkerke R² 0.217 for software volume against 0.099 for major axis length	72 adults, Urolithiasis 2021
Attenuation that makes disintegration less likely	Above 1,000 HU with high homogeneity	EAU urolithiasis guideline, 2026

A stone is reported as a length and behaves as a volume

Every report gives a stone a size, and that size is almost always a maximum diameter. Burden is a volume. The published approximation is the ellipsoid, π/6 times the product of three orthogonal diameters, and the consequence of the cube is the one thing worth taking away from this page: a 10 mm stone is eight times the burden of a 5 mm stone, and a 15 mm stone is twenty-seven times. A stone that has ‘only grown 2 mm’ from 8 to 10 mm has gained nearly twice its burden.

Three variants of the formula are in circulation and they do not agree. The scalene form uses all three diameters; the oblate form uses the longest twice and the shortest once; the prolate form uses the longest once and the second twice. In 72 adults treated with shockwave lithotripsy, all three correlated strongly with a software-segmented volume — the oblate and scalene forms above r = 0.9, the prolate above 0.8 — while giving significantly different volumes for the same stone. For the 10 × 7 × 5 mm stone used as the worked example here they differ by 43 per cent. All three are printed, so the choice is never silent. Note that only the scalene form is symmetric in its three inputs; the other two use one diameter twice, so which measurement goes in which field changes their answers.

The accuracy picture is worse than the spread between formulas suggests, and it is worth knowing before quoting the number. Against physically determined volumes, best-fit ellipsoid formulas overestimated by 27 per cent for stones under 9 mm and 89 per cent for stones of 20 mm and above. The linear measurements underneath carry their own error: three radiologists measuring the same stone differed by an average of 26.3 per cent in one series and by 1.2 to 1.9 mm in another, and that error is cubed here. So this is a burden index for comparing like with like, not a physical volume — and a stone measured on CT is not the same stone measured on a plain film.

The guideline position is deliberately unresolved. The EAU stratifies by cumulative linear diameter — under 5 mm, 5 to 10 mm, 10 to 20 mm, over 20 mm — while stating that volume predicts stone-free status better and that the absence of cut-off values guiding treatment is what keeps volume out of practice. That is why this page carries no bands: there is nothing published to band. The equivalent sphere diameter under the headline is the bridge, being the single diameter that carries the computed volume, and for the worked example it places a stone whose longest diameter is 10 mm into the 5 to 10 mm stratum instead. The 24-hour urine stone-risk panel and its interpretation already live in the urine chemistry category and are not rebuilt here. What a stone is made of, and how a CT attenuation number does and does not bear on that, is on the Hounsfield units page; how often a ureteric stone of a given size passed in the published series is on the spontaneous passage page.

Frequently asked questions

Why does an 8 mm stone matter so much more than a 5 mm one?

Because burden is a volume. Treated as spheres, 5 mm is 65 mm³ and 8 mm is 268 mm³ — four times as much stone for a 60 per cent increase in diameter. Doubling the diameter multiplies the burden by eight exactly.

Which ellipsoid formula should I use?

All three are published and all three are shown. The scalene form, π/6 × a × b × c, uses all the information you have and is the headline here. In the series that compared them, the oblate and scalene forms correlated with software-segmented volume above r = 0.9 and the prolate above 0.8, while giving significantly different volumes for the same stone — so record which one you used.

Is this the stone’s real volume?

No, and the error is large and in a known direction. Best-fit ellipsoid formulas overestimated physically determined volumes by 27 per cent under 9 mm and 89 per cent at 20 mm and above, and the linear measurements they rest on vary by about a quarter between observers. Use it to compare, not to quote.

Why are there no coloured bands on this page?

Because no volume strata have been published. The EAU says in terms that volume predicts stone-free status better than diameter and that the lack of cut-off values guiding treatment is why volume is not used clinically. Its own strata are linear — under 5 mm, 5 to 10 mm, 10 to 20 mm, over 20 mm — so the equivalent sphere diameter above is the number to read against them.

Where do I interpret a 24-hour urine stone-risk panel?

On the 24-hour urine stone-risk interpreter, which reads urine volume, calcium, oxalate, citrate, uric acid, sodium and pH against the EAU’s limits and checks the creatinine for collection adequacy first. The urine collection adequacy calculator and the six converters behind the panel are separate pages and are not duplicated here.

What this page will not tell me

Every threshold, range and performance figure on this page is the published figure from the source named beside it, and each one depends on the population, the method and the equipment it was derived in. Where your own report, laboratory or local guideline gives a different figure, that figure governs. A score, an index, a measured volume or an attenuation value is not a diagnosis, and a proportion measured in a cohort is not a probability for one patient. This page reports a figure and what the published sources attach to it. It does not make a clinical decision and cannot.

Related calculators

References

Cui HW, Tan TK, Christiansen FE, Osther PJS, Turney BW. The utility of automated volume analysis of renal stones before and after shockwave lithotripsy treatment. Urolithiasis. 2021;49(3):219–226.
The future of urolithiasis measurement: determining stone volume. AUANews. May 2024 — the published overestimates of the best-fit ellipsoid formulas and the interobserver error in linear measurement.
European Association of Urology. Guidelines on Urolithiasis, 2026 edition — Guidelines chapter, including the cumulative-diameter size strata, the spontaneous passage proportions in section 3.4.9.a and the shockwave-lithotripsy attenuation statement. uroweb.org.

Not medical advice. For healthcare professionals and education. Reference intervals vary by laboratory and assay — always use your own laboratory's. Never base a dose or a treatment decision on this page alone. Full disclaimer at calcengines.com/disclaimer/