How this instrument works
A cylinder's volume is base area times height, V = πr²h, and this sheet runs that identity backwards: divide the volume by the base area πr² and what is left over is exactly the height. Think of pouring a measured volume of liquid into a graduated cylinder of known radius — the liquid rises until the circular cross-section, area πr², times the level height, equals the poured amount. Reading off that level is arithmetically the same operation as h = V ⁄ (πr²).
Unlike a cone or a pyramid, a cylinder never tapers, so recovering its height needs no square root, no cube root, and no leading one-third — just a single division. Every horizontal slice through a cylinder has the identical cross-sectional area πr², so if you know how much space the whole solid encloses and how wide that unchanging cross-section is, the height falls straight out of one division. That directness is what makes the formula reliable for backfilling a missing spec: given a tank's rated capacity and the footprint it must fit, h = V ⁄ (πr²) hands back the one dimension the data sheet left blank.
The radius carries more weight than the volume does, because it is squared in the denominator. Double the volume at a fixed radius and the required height exactly doubles — but double the radius at a fixed volume and the required height falls to a quarter, since π(2r)² is four times πr². Push the radius toward zero and the height this formula demands grows without bound, and at r = 0 the expression is undefined outright: a cylinder with no width has no cross-section to fill, so no finite height could ever hold a positive volume inside it.
- 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 radius into the Radius field, using the matching linear unit — metres alongside cubic metres, feet alongside cubic feet.
- Read Height: the sheet computes h = V ⁄ (πr²) the instant both fields hold a value, and updates live if either one changes.
- Check the answer by multiplying π × Radius² × Height — the product should reproduce your original Volume figure.
Worked example — volume 100, radius 5
A cylindrical rainwater tank needs to hold a volume of 100 cubic metres and the site only leaves room for a radius of 5 metres. Squaring the radius gives r² = 25, and multiplying by π puts the base area at 78.53981633974483 square metres. Dividing the required volume by that base area, h = 100 ⁄ 78.53981633974483 = 1.2732395447351628 metres — call it 1.27 m, the wall height the fabricator needs to cut the tank's steel to.
Running the volume formula forward confirms the inversion: π × 5² × 1.2732395447351628 = π × 25 × 1.2732395447351628 = 100 exactly, the round trip that catches a mistyped radius before a tank gets built too short or too tall.
Questions
What is the formula for the height of a cylinder?
h = V ⁄ (πr²), found by dividing a cylinder's volume by the area of its circular base. It is V = πr²h solved for h: divide both sides by πr² and the height is what remains. For a volume of 100 and a radius of 5, that gives h = 100 ⁄ 78.53981634 ≈ 1.2732 — a single division, with no root of any kind involved.
How is this different from finding the height of a cone or a square pyramid?
Those two taper to a point, so their volume formulas carry a leading one-third and their height formulas need a square root to undo a squared term. A cylinder has the same circular cross-section all the way up, so its volume formula has no one-third and its height formula, h = V ⁄ (πr²), needs no root at all — just division by the base area.
Why does doubling the radius shrink the required height instead of growing it?
Because radius sits squared in the denominator: h = V ⁄ (πr²). Holding volume fixed and doubling the radius multiplies πr² by four, so the height needed to hold that same volume drops to a quarter, not a half. A wide tank only needs to be a quarter as tall as a narrow one holding the identical volume, at double the radius.
What happens to the height formula when the radius is zero?
It is undefined — division by zero. A cylinder with zero radius has no circular cross-section at all, so no finite height, however tall, could enclose a positive volume inside it. Physically this matches intuition: a line segment has no room to hold water, no matter its length.
How can I check that a computed height is correct?
Multiply π × Radius² × Height and confirm it reproduces the original Volume. For radius 5 and the computed height 1.2732395447351628, π × 25 × 1.2732395447351628 returns 100 exactly, the same figure the tank's capacity spec started with.
Does the formula work in any consistent unit system?
Yes — enter Volume in cubic units and Radius in the matching linear unit, cubic metres with metres or cubic feet with feet, and Height returns in that same linear unit. Mixing systems, say cubic metres with a radius given in feet, produces a height figure with no physical meaning.