What a frustum is
Cut the tip off a cone with a slice parallel to its base, and the piece left is a frustum (from the Latin for “piece”). Many everyday objects are frustums: buckets, flowerpots, drinking cups, lampshades, funnels and the bases of some columns.
The formulas
With bottom radius R, top radius r and height h:
- Volume: V = π × h × (R² + R × r + r²) ÷ 3
- Slant height: s = √(h² + (R − r)²)
- Lateral (side) area: A = π × (R + r) × s
- Total area: A + π × R² + π × r²
The volume formula was known to Egyptian mathematicians: the Moscow papyrus, from around 1850 BC, computes the volume of a square-based frustum of a pyramid with the same pattern.
The calculator takes diameters, since they are easier to measure, and halves them.
Worked example
A bucket measuring 10 inches across the bottom, 12 inches across the top and 14 inches tall (inside):
| Result | Value |
|---|---|
| Volume | 5.78 gallons (1,334 cubic inches) |
| Slant height | 14.04 in |
| Side surface area | 485 sq in |
| Total surface area | 677 sq in |
| Side angle from vertical | 4.1° |
The total area counts both ends; for an open bucket, subtract the top circle (π × 6² ≈ 113 sq in).
Practical uses
- Planters and buckets: find the capacity in gallons or liters to buy the right amount of potting soil.
- Lampshades and cone patterns: the slant height and radii define the flat pattern you cut from fabric or sheet metal.
- Concrete and earthworks: tapered footings and round pier forms are frustums.
- Cups and containers: check whether a cup really holds what its label says.
Checks
If the top and bottom diameters are equal, the frustum becomes a cylinder, and the formula reduces to π × r² × h. If the top diameter is zero, it becomes a full cone, π × R² × h ÷ 3. The average-area shortcut, h × (area of top + area of bottom) ÷ 2, overestimates the volume; the true frustum formula is always slightly smaller.