Frustum Calculator: How to Find the Volume and Surface Area of a Truncated Cone
About the Author
Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
A frustum calculator answers a question hiding inside half the objects in your kitchen: how much does a tapered container actually hold? Buckets, flower pots, paper cups, lampshades, pint glasses — none of them are true cylinders. They’re frustums: cones with the pointed top sliced off by a cut parallel to the base. Feed the tool above two radii and a height and it returns the volume, slant height, and every surface area at once, with the arithmetic written out line by line. What follows is the reference material to go with it — the three formulas that do all the work, a real bucket measured to the millilitre, the reason the “just average the radii” shortcut always comes up short, and the sheet-metal trick that unrolls a truncated cone into a flat pattern you can cut.
The Formulas a Frustum Calculator Actually Runs
Call the bottom radius R, the top radius r, and the perpendicular distance between the two circular ends h. Every result a conical frustum calculator produces comes from these four lines:
V = (π · h / 3) · (R² + R·r + r²)
s = √((R − r)² + h²)
Lateral area = π · (R + r) · s
Total area = π · (R + r) · s + π·R² + π·r²
Two things make these formulas forgiving. First, R and r are interchangeable — swap them and nothing changes, so it doesn’t matter whether your shape is wider at the top (a bucket) or at the bottom (a lampshade). Second, they degrade gracefully: set r = 0 and the volume collapses to πR²h/3, the ordinary cone. Set r = R and it becomes πR²h, the cylinder. A quick trial run with R = 6, r = 3, h = 8 gives s = √(9 + 64) = 8.544, and V = (π · 8/3)(36 + 18 + 9) = 168π ≈ 527.8 cubic units. Keep those numbers in mind — the same shape reappears two sections down, built a completely different way.
One practical warning before the examples: real objects get measured rim-to-rim with a tape, which gives diameters, not radii. Halve them first, or use the calculator’s diameter mode and let it do the halving. Feeding diameters straight into the radius formula inflates the volume by a factor of four.
Your Bucket Is a Frustum — Here’s What It Holds
Take an ordinary household bucket: 28 cm across the opening, 22 cm across the base, 25 cm tall inside. Radii of 14 and 11. The volume calculation runs:
R² + R·r + r² = 196 + 154 + 121 = 471
V = (π · 25 / 3) · 471 = 3,925π ≈ 12,331 cm³
Since 1,000 cm³ is exactly one litre, that bucket holds about 12.3 litres to the brim — 3.26 US gallons. Notice the label on a bucket like this would probably say 10 L. Manufacturers rate capacity to a fill line a few centimetres below the rim, so a frustum volume calculator will consistently report more than the printed number. Neither figure is wrong; they measure different things. The same brim-versus-line gap shows up in measuring jugs, mixing tubs, and tapered pint glasses, and it’s worth remembering any time you use a volume calculator to check a container against its label.
The slant height of this bucket, for completeness: s = √((14 − 11)² + 25²) = √634 ≈ 25.18 cm. Barely longer than the vertical height, because the taper is gentle — a point that matters a great deal in the section after next.
Where R² + R·r + r² Comes From
The middle term R·r looks odd until you rebuild the frustum the way geometry actually makes one: start with a full cone, slice off the top. Extend the sloped sides of any frustum and they meet at an apex. The full cone’s height works out to H = h·R/(R − r), and the sliced-off tip is itself a smaller cone of height H − h = h·r/(R − r). Subtract the tip’s volume from the whole and the frustum is what remains.
Run that on the R = 6, r = 3, h = 8 shape from earlier. The apex sits at H = 8 × 6/(6 − 3) = 16, so the removed tip is a cone of height 8 and radius 3. Whole cone: (π/3) · 36 · 16 = 192π. Tip: (π/3) · 9 · 8 = 24π. Difference: 168π — exactly what the direct formula gave. The algebra behind the shortcut is the difference of cubes: (R³ − r³)/(R − r) = R² + R·r + r², which is where that middle term sneaks in. Wolfram MathWorld’s conical frustum entry walks through the full derivation with the integrals on display, and our cone calculator handles the un-truncated case whenever the two-cone subtraction is the way you’d rather think about it.
The Averaging Shortcut Undervalues Every Frustum
There’s a tempting mental shortcut: treat the truncated cone as a cylinder whose radius is the average of the two ends, (R + r)/2. It feels like it should work. It never does — and it always errs in the same direction. Compare the two expressions: the true formula uses (R² + R·r + r²)/3, the shortcut uses ((R + r)/2)². Subtract one from the other and the gap is exactly (R − r)²/12, so the missing volume is:
V_true − V_averaged = π · h · (R − r)² / 12
A squared term can’t be negative, so the cylinder estimate is always an undercount, and the penalty grows with the square of the taper. How bad it gets in practice:
| Object | R / r / h | True volume | Averaged estimate | Shortfall |
|---|---|---|---|---|
| Household bucket | 14 / 11 / 25 | 12,331 cm³ | 12,272 cm³ | 0.5% |
| Paper cup | 4 / 2.5 / 9 | 304 cm³ | 299 cm³ | 1.7% |
| Wide garden funnel | 15 / 5 / 20 | 6,807 cm³ | 6,283 cm³ | 7.7% |
| Full cone (r = 0) | 6 / 0 / 12 | 452 cm³ | 339 cm³ | 25% |
The pattern is worth internalising. On a gently tapered bucket the shortcut is off by half a percent — harmless. On a cone it’s off by a quarter of the whole volume, and 25% is the worst case, reached exactly when one radius hits zero. So the shortcut is a decent sanity check but a bad substitute: if your estimate and the true frustum answer disagree by more than a few percent, the shape tapers strongly and the cylinder model has broken down. For genuinely straight-sided containers, of course, the cylinder calculator is the right tool and no correction term exists to worry about.
Slant Height, Lampshades, and Flat Patterns
Volume only ever uses the vertical height h. The moment you care about the outside of the shape — fabric for a lampshade, steel for a chute, paper for a cup sleeve — the slant height s takes over, and the two are easy to confuse when measuring. A tape laid along the sloped side of the object gives you s. Measuring straight up beside it gives h. Since s = √((R − r)² + h²), s is always the longer of the two, and using one where the other belongs shifts every downstream number.
A concrete case: a drum lampshade frame with a 36 cm bottom diameter, 20 cm top diameter, and 24 cm vertical height. So R = 18, r = 10, and s = √(8² + 24²) = √640 ≈ 25.30 cm. The fabric needed is the lateral surface: π(R + r)s = π × 28 × 25.30 ≈ 2,225 cm², about 0.22 m² before seam allowance. The end circles don’t count here — a shade has no top or bottom panel — which is exactly why the calculator reports the lateral piece separately from the figure a general surface area calculator would give for the closed solid.
Now the part most tutorials skip: what shape do you actually cut? Unrolled flat, a frustum’s side is not a rectangle and not a triangle — it’s an annular sector, a bite taken out of a washer. Its two arc radii are the straight-line distances from the cone’s apex to each rim, L = s·R/(R − r) on the outside and L − s on the inside, and the sector spans 360° × (R − r)/s. For the lampshade: outer radius 25.30 × 18/8 ≈ 56.9 cm, inner radius 31.6 cm, sector angle 360 × 8/25.30 ≈ 114°. Draw two arcs with a string compass, connect the ends, and the piece wraps perfectly around the frame. Sheet-metal workers call this a cone development, and it’s the same layout whether the material is silk, galvanised steel, or cardboard.
Square Frustums and a 3,850-Year-Old Math Problem
Nothing about truncation is special to circles. Slice the top off a square pyramid and the same logic produces a square frustum with volume V = (h/3)(a² + a·b + b²), where a and b are the two edge lengths — the circular formula with π stripped out and radii swapped for sides. The fully general version works for any two similar parallel faces: V = (h/3)(A₁ + A₂ + √(A₁·A₂)), taking the two end areas directly. Our pyramid calculator covers the untruncated square case when you need the full solid instead.
This is also the oldest volume formula we can prove humans knew. Problem 14 of the Moscow Mathematical Papyrus, an Egyptian scroll dating to roughly 1850 BC, computes the volume of a truncated pyramid with base edge 4, top edge 2, and height 6. The scribe’s recipe: square the 4 to get 16, multiply 4 by 2 to get 8, square the 2 to get 4, add them to 28, take a third of the height — 2 — and multiply. Answer: 56, and it’s exact. That is precisely (h/3)(a² + ab + b²) executed step by step, more than a millennium before Greek geometry got around to proving why it works. When you punch numbers into a frustum calculator, you’re running what may be the single oldest nontrivial algorithm still in daily use — the arithmetic hasn’t changed, only the speed.
