Truncated Cone Volume (Conical Frustum)

A truncated cone (conical frustum) is what you get when you slice off the tip of a cone parallel to its base. Buckets, paper coffee cups, lampshades, and many storage tanks have this shape.

V = (1/3) × π × h × (R² + r² + R × r)

Where R is the bottom (larger) radius, r is the top (smaller) radius, and h is the vertical height between the two parallel circles.

Worked example — 5-gallon plastic bucket: A standard contractor’s bucket has R = 5.875 in (bottom inside), r = 6.75 in (top inside), h = 14 in. Wait — buckets taper outward going up! So R (bottom) is the smaller one. Let me use the convention: R = larger end, r = smaller end. For this bucket: R = 6.75, r = 5.875, h = 14. V = (1/3) × π × 14 × (45.56 + 34.52 + 39.66) = (1/3) × π × 14 × 119.74 ≈ 1,755 in³ ≈ 7.6 US gallons of inner volume.

Hold on — a “5-gallon” bucket holds 7.6 gallons? Yes. Industry “5-gallon buckets” are designed to fit 5 gallons comfortably with no risk of spillage. The remaining 2.6 gallons is freeboard above the fill line.

Where truncated cones matter:

• Plastic buckets and pails. All the standard household and industrial pails are truncated cones, slightly wider at the top for stack-nesting.
• Lampshades. Cylindrical lampshades are rare; most taper.
• Paper coffee cups. Hot-drink cups taper to a smaller base for thermal grip.
• Conical hopper bottoms. Silos and bins with bottom hoppers — the hopper is a truncated cone if it has a flat bottom outlet.
• Champagne flutes. Mostly truncated-cone-shaped (with various tweaks).
• Lawn-care chemical containers. Many granular fertilizer scoops use a truncated-cone shape.

Why the formula has all three of R², r², and R×r:

The truncated cone is the “average” of a cylinder (R = r) and a complete cone (r = 0). The R²+r²+R×r term smoothly interpolates between them.

Two useful limits:

• If R = r (no taper): V = (1/3) × π × h × 3R² = π × R² × h, which is the cylinder formula. ✓
• If r = 0 (complete cone, no truncation): V = (1/3) × π × h × R², which is the cone formula. ✓

Alternative form using the slant height (sometimes simpler if you measure the slant directly): V = (1/3) × π × h × (R² + r² + R × r) — same formula, just emphasizes that h is the perpendicular (vertical) height, not the slant.

Sanity check:

• h = 0: V = 0. ✓
• R = r = 0: V = 0. ✓
• R, r, h all equal: V = (1/3) × π × R × 3R² = π × R³.

Pro tip — for capacity at intermediate fill heights: If you’re trying to figure out “how full is this bucket when I’ve added 3 liters?”, you can’t just divide by total volume. The lower section is narrower and holds less per inch than the upper section. Use the formula with R and r at the actual fill level — interpolate linearly between the bottom and top radii based on fill height fraction.