How to Find the Frustum Volume of Any Bucket, Lampshade or Hopper

Pick up almost any bucket in your house and look at it from the side. The rim is wider than the base. That taper is not an accident — it lets buckets stack inside each other, pour without glugging, and shed rainwater. The shape has a name: a frustum. It is what you get when you take a cone and slice the pointy tip off with a cut parallel to the base. Drinking cups, terracotta flower pots, lampshades, grain hoppers, and traffic-cone bottoms are all frustums too.

The trouble is that the volume of a frustum is one of those formulas that looks fussy enough that most people reach for a guess instead. A 19-litre pail does not actually hold 19 litres of water to the brim, and "average the top and bottom and multiply by height" gives the wrong answer. Here is how the real formula works, why it has the shape it does, and how to get a trustworthy number in seconds.

What a frustum actually is

A cone tapers smoothly from a circular base up to a single point. Chop that point off with a flat cut parallel to the base, and the leftover lower chunk is a conical frustum. It has two parallel circular faces — a bigger one and a smaller one — joined by a slanted side.

Three measurements fully describe it:

R, the radius of the larger circle (usually the base, or the wide rim of a bucket)
r, the radius of the smaller circle (the narrow end)
h, the perpendicular height — the straight up-and-down distance between the two flat faces

Notice that h is not the slanted side. The sloped edge is longer than h, and confusing the two is the single most common mistake people make. More on that below.

The frustum volume formula

For a conical frustum the volume is:

V = (1/3) π h (R² + R·r + r²)

Read it slowly. You square the big radius, you square the small radius, and crucially you add a middle term, R·r, which is the two radii multiplied together. That cross term is the whole point. If you drop it and write (1/3)πh(R² + r²), you undercount every frustum you ever measure. The R·r term is what makes a frustum more than just the crude average of two cones — it accounts for the smooth taper of the wall in between.

Two quick sanity checks show how well-behaved this formula is:

Set r = 0 (the small circle shrinks to a point) and the formula collapses to (1/3)πR²h — the volume of an ordinary cone.
Set r = R (no taper at all) and it becomes πR²h — a plain cylinder.

So one equation quietly covers cones, cylinders, and everything between them. If you only need the un-cut cone case, the dedicated cone volume calculator handles that limiting shape directly.

A worked example: how much does that bucket really hold?

Let me run a real bucket through it, because this is the case I check most often myself. I keep a galvanized pail in the garage that I use for mixing wall filler, and I had no idea of its honest capacity until I measured it. The inside top radius came out to R = 15 cm, the inside bottom radius to r = 12 cm, and the inside height to h = 30 cm.

Plug those in:

V = (1/3) · π · 30 · (15² + 15·12 + 12²)
  = (1/3) · π · 30 · (225 + 180 + 144)
  = (1/3) · π · 30 · 549
  = 10 · π · 549
  = 5490 π
  ≈ 17 247 cubic centimetres

Since 1 litre is 1000 cubic centimetres, that pail holds about 17.2 litres — not the round 20 litres I would have guessed from its size. That gap matters when you are buying paint by the litre, mixing a precise ratio, or working out how many trips to the tap a job will take.

It is worth pausing on the middle term again. If I had carelessly used (1/3)πh(R² + r²) and skipped the R·r, I would have computed (1/3)·π·30·(225 + 144) = 3690π ≈ 11 592 cm³, roughly 11.6 litres. That is a third short of reality — a serious error for something as simple as a bucket.

Vertical height versus slant height

The other classic trap is the slanted side. The sloped wall of the frustum has its own length, called the slant height:

slant ℓ = √( h² + (R − r)² )

For my bucket that is √(30² + 3²) = √909 ≈ 30.15 cm. It is barely longer than h here because the taper is gentle, but on a steep funnel the difference is dramatic.

The rule is simple: use the vertical height h for volume, and the slant height ℓ for the side surface area. Volume is about how much space is enclosed, which is a straight-up-and-down measurement. Surface area is about the material that wraps the slope, so it follows the slanted line. If you are cutting sheet metal for a hopper, the lateral (side) area you need is π(R + r)·ℓ, and getting ℓ wrong wastes metal.

Where frustums show up in real life

Once you know the shape, you start seeing it everywhere:

Buckets and pails. Wider at the top so they nest and pour. Measure inside dimensions for true capacity.
Drinking cups and paper cups. The slight taper is structural and lets them stack at the counter.
Lampshades. Often you want the lateral area to estimate how much fabric or paper covers the frame, not the volume.
Flower pots. Volume tells you how much potting soil a pot swallows.
Hoppers, funnels and chimney transitions. Open at both ends, so you usually want lateral surface area and slant height rather than enclosed volume.
Storage silos and grain bins. The conical lower section that funnels material out is a frustum.

For any of these you also need consistent units — measure everything in centimetres and read litres, or measure in inches and read cubic inches. Mixing inches and feet across the three inputs throws the answer off by a factor of twelve, a mistake that is surprisingly easy to make on a tape measure.

Skip the arithmetic

The formula is not hard, but doing it by hand invites slips: a dropped cross term, a confused height, a stray unit. When I want a number I trust without re-checking my own multiplication, I punch the three values straight into the Frustum Volume Calculator. It returns the volume, the slant height, both circle areas, the lateral surface area, and the total surface area at once, and it runs entirely in the browser so nothing you measure leaves the page. Set the top radius to zero and it gives you a cone; set the two radii equal and it gives you a cylinder — the same one formula, covering the whole family of shapes you actually meet.

Made by Toolora · Updated 2026-06-13