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: a 5-gallon plastic bucket

Note the convention first, because a bucket is the classic trap. A pail is WIDER AT THE TOP, so its top radius is R (the larger one) and its bottom radius is r. Get those the wrong way round and the answer is still valid arithmetic for a differently shaped object, which is why nothing looks wrong.

A common contractor’s pail measures roughly 11.8 in across inside the rim and 10.4 in across the base, standing about 13.5 in to the brim. That gives R = 5.9, r = 5.2, h = 13.5 in.

V = (1/3) × π × 13.5 × (34.81 + 27.04 + 30.68) = (1/3) × π × 13.5 × 92.53 ≈ 1,308 in³

At 231 cubic inches to the US gallon that is 5.66 gallons brim-full, so the 5-gallon rating leaves about half a gallon of freeboard. That is the usual arrangement: the badge is the working capacity, not the geometric one, and the gap is small.

Measure your own bucket before trusting any of these numbers. Pail dimensions vary by manufacturer and the inside is what counts, not the outside, since the wall and the ribs are worth a surprising amount over 13 inches of height.

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, where the hopper is a truncated cone whenever it ends in a flat outlet rather than a point.
• 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. ✓

If you measured the slant instead of the height

The h in the formula is the VERTICAL height, the perpendicular distance between the two circles. A tape run down the outside gives you the slant l, which is longer. Convert before you compute:

h = √(l² − (R − r)²)

On the bucket above, l = √(0.7² + 13.5²) = 13.52 in, barely different from h because the taper is slight. On a steeply flared shape the gap is large, and using a slant as though it were a height overstates the volume every time.

Sanity check:

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

Capacity at intermediate fill heights

You cannot pro-rate. Fill a bucket to half its height and it holds more than half its volume, because the wide part is at the top. Work out the radius at the fill line first, interpolating between the two ends:

r_fill = r + (R − r) × (fill height / total height)

then apply the same formula to the shorter frustum from the base up to there. On the bucket above, filling to half height gives r_fill = 5.55 in and 613 in³, which is 47% of the total, not 50%. Turn the bucket upside down and the same half-height fill would be 53%.