How this instrument works
V = ⅓πr²h is the cone's volume identity; multiply by three, divide by π times the height, and take a square root of what remains, and the result is r = √(3V ⁄ πh) — the same statement rearranged for its other unknown. The root is unavoidable because radius carries the square in the original identity: freeing a squared term always means undoing a square somewhere, exactly as a square's area only gives back a side length after a root is taken. The companion sheet on this site that solves the same identity for height needs no root at all, since height sits in the formula to the first power only.
Because the map from volume to radius runs through a square root, small errors shrink rather than grow as they pass through it: a volume reading that is 10% too high yields a radius only about 4.9% too high, since √1.10 ≈ 1.049. That compression of relative error is a general feature of square-root formulas, not a coincidence of these particular numbers — doubling the measured volume outright multiplies the recovered radius by only √2 ≈ 1.414, not by two, so a true doubling of radius demands a fourfold increase in volume, not a twofold one.
The formula behaves exactly as the shape does at its limits. Let the height shrink toward zero while the volume stays fixed and the required radius grows without bound, since πh sits in the denominator — a nearly flat cone can only trap real volume by spreading its base absurdly wide. Let the volume itself shrink toward zero instead, at any height, and the radius collapses to zero as well: a cone enclosing nothing has degenerated into a vertical line segment. Between those extremes sits the ordinary case this instrument targets — a grain hopper, a paper filter, or a conical tank where a fill gauge gives the volume and a tape gives the height, but the base is buried out of reach of direct measurement.
- Enter the cone's Volume — its capacity, in whatever cubic unit your other figures already use.
- Enter Height — the straight-up rise from base to apex, not the slanted side, in a matching length unit.
- Read Radius for the result of r = √(3V ⁄ πh), the distance from the base's centre out to its rim.
- To check by hand, square the Radius shown, multiply by π and by Height, then divide by three; the figure that comes out ought to match the Volume you entered, within rounding.
Worked example — a 100-unit funnel standing 12 units tall
A conical funnel holds Volume = 100 cubic units and stands Height = 12 units from its spout to its rim. Three times the volume is 300; π times the height is 37.69911184307752; dividing the two leaves 7.957747154594767 under the root, and the square root of that quotient is Radius = 2.8209479177387813 — exactly what this instrument returns for those two inputs.
Running the identity forward confirms it without rounding drama: squaring 2.8209479177387813 returns 7.957747154594766, and one third of π times that squared radius times the Height of 12 comes back to 99.99999999999999 — the original volume, recovered to within a sliver of floating-point rounding. The π introduced when the radius gets squared cancels the π divided out to find it, so the round trip is exact in principle even though a square root sat in the middle of it.
Questions
What is the formula for finding a cone's radius from its volume and height?
r = √(3V ⁄ πh), the volume identity V = ⅓πr²h solved for radius instead of height. Triple the volume, divide by π times the perpendicular height, then take the square root — using volume 100 with height 12 that works out to r = √(300 ⁄ 37.699111...) ≈ 2.820947918.
Why does isolating the radius need a square root when isolating the height doesn't?
The identity V = ⅓πr²h keeps radius squared but leaves height as a lone, first-power factor. Undoing a lone factor is a plain division — the companion height-of-cone sheet on this site does exactly that — while undoing a squared factor means reversing a square, which is where the root in r = √(3V ⁄ πh) comes from.
How much does an error in the measured volume affect the computed radius?
Roughly half as much, proportionally. Because radius is volume raised to the one-half power at fixed height, a volume reading 21% too high yields a radius exactly 10% too high (√1.21 = 1.1), and a volume 10% too high yields a radius only about 4.9% too high. Square roots compress relative error; they never amplify it.
What's the most common mistake when using this formula?
Entering the slant height in place of the perpendicular height. The h in r = √(3V ⁄ πh) is the straight vertical distance from apex to the base's plane, not the longer distance measured down the cone's sloped side; using the slant figure overstates h and understates the radius the formula returns.
If the volume doubles, does the radius double too?
No — at a fixed height, radius tracks the square root of volume, not volume itself. Doubling Volume multiplies Radius by only √2 ≈ 1.414; getting a genuinely doubled radius takes four times the original volume, because volume depends on r squared rather than on r alone.
How does this differ from finding the radius of a cylinder or a sphere?
Each solid has its own volume formula, so each rearrangement differs. A cylinder's V = πr²h gives r = √(V ⁄ πh) with no factor of three; a sphere's V = (4⁄3)πr³ gives r = ∛(3V ⁄ 4π), a cube root rather than a square root, since the sphere's radius is cubed rather than squared.