a frustum
/ FRUSS-tum /
Take a cone or a pyramid and slice off its pointed top with a cut parallel to the base. What remains — the chunk between the base and the slice, like a lampshade, a bucket, or a paper coffee cup — is a frustum. The word is Latin for a piece broken off; both the bucket shape and the truncated pyramid of a building's base are everyday frustums.
A frustum has two parallel faces of different sizes (the original base and the smaller cut face) joined by a slanted band. The cleanest way to find its volume is by subtraction: take the volume of the whole original cone or pyramid and subtract the volume of the small cone or pyramid that was sliced off the top, both using V = (1/3) · B · h. For a conical frustum with base radii R and r and height h there is also a direct formula, V = (1/3) · pi · h · (R^2 + R·r + r^2). The lateral (slanted) surface of a conical frustum is pi · (R + r) · L, where L is the frustum's slant height.
The most common error is to treat a frustum as if it were a simple prism or cylinder of some 'average' radius — it is not, and that shortcut gives the wrong volume because the cross-section shrinks nonlinearly as you go up. When the cut is at the very apex you recover the full cone; when the cut is at the base you get a flat disk of zero height — the frustum interpolates between these extremes.
A bucket (conical frustum) with bottom radius 5 cm, top radius 8 cm, and height 12 cm has volume (1/3)·pi·12·(8^2 + 8·5 + 5^2) = 4·pi·(64 + 40 + 25) = 4·pi·129 = 516·pi ≈ 1621 cm^3.
A frustum is a cone or pyramid with its tip cut off, parallel to the base.
A frustum is not a cylinder of average radius; the cross-section changes nonlinearly, so use the subtraction method or the proper R^2 + R·r + r^2 formula.