How this instrument works
A truncated cone, or frustum, is what remains of a full cone once a smaller, similar cone is sliced off its top, leaving two parallel circular faces of different sizes joined by a slanted side. Its volume can be found the long way, by computing the full cone's volume and the sliced-off tip's volume separately and subtracting one from the other, but the direct formula V = πh(R² + Rr + r²) ⁄ 3 folds that subtraction in already, so no intermediate cone volume ever needs to be computed by hand.
The R² + Rr + r² term is exactly what survives after that subtraction algebraically, and it has two useful properties worth noticing: it treats the two radii symmetrically (swapping which one is called R and which is called r changes nothing), and it correctly reduces to simpler, already-familiar formulas at its two extremes. When the top radius r shrinks to zero, the frustum becomes an ordinary full cone, and the formula reduces to the standard ⅓πR²h. When the two radii are equal, the frustum's slanted sides straighten into a plain cylinder, and the formula reduces to πr²h.
Frustum shapes are common well beyond geometry class: buckets, lampshades, drinking cups, and the base sections of some silos and towers are all frustums, chosen specifically because a shape wider at one end than the other stacks, nests, or pours more conveniently than a plain cylinder.
- Enter the larger, bottom radius into the Bottom radius field.
- Enter the smaller, top radius into the Top radius field.
- Enter the perpendicular distance between the two circular faces into the Height field.
- Read Volume: the sheet applies πh(R² + Rr + r²) ⁄ 3 directly, with no separate cone-subtraction step.
Worked example — radii 5 and 3, height 4
A frustum has a bottom radius of 5, a top radius of 3, and a height of 4. The volume is V = π × 4 × (25 + 15 + 9) ⁄ 3 = π × 4 × 49 ⁄ 3 ≈ 205.25 cubic units — reached in one formula, with no separate full-cone or sliced-tip volume ever computed along the way.
Set the top radius to match the bottom one, 5 and 5 with the same height of 4: the formula gives π × 4 × (25 + 25 + 25) ⁄ 3 = 100π ≈ 314.16 cubic units, exactly the volume of a plain cylinder of radius 5 and height 4 — the frustum formula correctly collapsing to the cylinder formula once both radii match.
Questions
What is the formula for the volume of a truncated cone?
V = πh(R² + Rr + r²) ⁄ 3, where R and r are the two circular radii (larger and smaller) and h is the perpendicular height between them. This single formula already accounts for the subtraction between a full cone and the smaller cone sliced off its top.
Does it matter which radius I call R and which I call r?
The formula's R² + Rr + r² term is symmetric in R and r, so swapping the two labels doesn't change the volume. It is only convention, not a requirement of the math itself, that R usually refers to the larger, bottom radius.
What happens if the top radius is zero?
The frustum becomes an ordinary full cone, since the smaller circle has shrunk to a single point, and the formula reduces exactly to the standard cone volume formula, ⅓πR²h — the r and Rr terms both vanish, leaving only the R² term.
What happens if the two radii are equal?
The frustum's slanted side straightens out entirely, turning the shape into a plain cylinder, and the formula reduces to πr²h — the ordinary cylinder volume formula, since R² + Rr + r² becomes 3r² and dividing by 3 leaves just r².
How is this related to the surface area of a frustum?
They are separate calculations describing different things: this page returns the enclosed VOLUME, while a frustum's surface area (the lateral slanted surface plus the two circular caps) requires the slant height as an additional input and a different formula entirely, since surface area and volume never share a common derivation.