How this instrument works
A cylinder's volume is base area times height, V = πr²h, and the radius sits inside that base area as a squared term. Freeing it takes two moves: divide the volume by the height to strip away the one factor that isn't squared, leaving πr² = V ⁄ h, then divide by π and take the square root to undo the square. That is why the formula reads r = √(V ⁄ πh) rather than a plain fraction — the root is not decoration, it is the direct algebraic cost of r having been squared in the first place.
That single detail is what separates this page from its companion, the height-of-cylinder calculator, which solves the identical equation for the other unknown. Height appears only to the first power in V = πr²h, so isolating it is one clean division with no root anywhere. Radius appears squared, so isolating it needs that extra square-root step. Same identity, same three quantities, but which variable you solve for decides whether the arithmetic stays linear or grows a root — and it is worth naming the specific solid, too: a cone's volume carries a leading one-third that a cylinder's does not, so a cone's radius formula is r = √(3V ⁄ πh), not this one, and a sphere's volume has no height term at all, so its radius comes from a cube root, not a square root.
The formula is well-behaved everywhere height is positive, but it breaks down as height shrinks toward zero: to hold a fixed volume in an ever-shallower cylinder, the base has to spread ever wider, and r climbs without bound as h approaches 0 — at h = 0 exactly, the expression divides by zero and no finite radius could enclose any volume in a shape with no height at all. Only the positive square root is taken, since a negative radius describes no physical circle; the algebra technically admits ±√(V ⁄ πh), but the base's geometry throws the negative branch away before it ever reaches the answer.
- Enter the cylinder's known capacity into the Volume field, in cubic units — cubic metres, cubic feet, whatever the tank's spec sheet uses.
- Enter the fixed Height into the Height field, using the matching linear unit — metres alongside cubic metres, feet alongside cubic feet.
- Read Radius: the sheet computes r = √(V ⁄ πh) the instant both fields hold a value, and updates live if either one changes.
- Check the answer by squaring Radius, multiplying by π, then by Height — the product should reproduce your original Volume figure.
Worked example — volume 100, height 8
A cylindrical tank needs to hold a volume of 100 cubic metres and the site fixes its height at 8 metres — a low, wide footing rather than a tall column. Multiplying π by the height gives πh = 25.132741228718345, and dividing the required volume by that figure gives r² = 100 ⁄ 25.132741228718345 = 3.9788735772973833 square metres. Taking the square root, r = 1.9947114020071635 metres — call it 1.995 m, the base radius the fabricator needs to roll the tank's shell to, or about 3.989 m across if a diameter is what the doorway needs checking against.
Running the volume formula forward confirms the inversion exactly: π × (1.9947114020071635)² × 8 returns 100.0 with no rounding residue at all, the round trip that catches a mistyped height before a tank is welded the wrong width. Compare this to the height-of-cylinder page's own worked example, volume 100 with radius 5 giving height 1.2732395447351628 — same identity, opposite unknown, and a square root only on this side of it.
Questions
What is the formula for the radius of a cylinder given its volume and height?
r = √(V ⁄ πh), found by dividing the volume by π times the height, then taking the square root. It is V = πr²h rearranged to isolate r: divide both sides by πh to reach r² = V ⁄ πh, then root both sides. For a volume of 100 and a height of 8, that gives r = √3.9788735772973833 ≈ 1.9947.
How does this differ from the height-of-cylinder calculator, which solves the same equation?
Height appears only to the first power in V = πr²h, so the height-of-cylinder page isolates it with a single division and no root. Radius appears squared, so freeing it here needs an extra step — dividing by πh and then taking a square root — because undoing a square always costs a root that undoing a linear term does not.
Why isn't this the same formula as the radius-of-cone or radius-of-sphere calculators?
Each solid has its own volume formula. A cone's carries a leading one-third, so its radius formula is r = √(3V ⁄ πh) — a factor of three heavier under the root than a cylinder's. A sphere's volume, ⁴⁄₃πr³, has no height term at all, so its radius comes from a cube root, not a square root. Naming the solid changes the arithmetic, not just the label.
What happens to the required radius as the height shrinks toward zero?
It grows without bound. Since height sits in the denominator under the root, an ever-shallower cylinder needs an ever-wider base to hold the same volume, and the formula is undefined at h = 0 — a cylinder with zero height has no room to enclose any volume, however wide its circular face.
How can I check that a computed radius is correct?
Square the radius, multiply by π, then multiply by the height, and confirm it reproduces the original volume. For height 8 and the computed radius 1.9947114020071635, π × (1.9947114020071635)² × 8 returns 100.0, the same figure the tank's capacity spec started with.
Which unit should Volume and Height be entered in?
Any unit system works as long as it's consistent — cubic metres with metres, cubic feet with feet — and Radius comes back in that same linear unit. Mixing systems, say a volume in litres against a height measured in feet, produces a radius figure with no physical meaning.