A hopper-bottom digester isn't one shape — it's a cylinder sitting on top of a cone, and if you calculate its volume as if it were a plain cylinder, you'll overestimate capacity and misjudge detention time. This guide covers the three-dimensional shapes and decomposition techniques operators need for irregular, real-world structures: cones, frustums, spheres, and composite shapes built from simpler pieces.
Volume of a Cone
Hopper-bottom tanks, digester bottoms, and sludge collection hoppers taper to a point (or near-point) — that tapered shape is a cone.
Volumecone=31×πr2×h Notice this is exactly one-third of the cylinder volume formula (πr2h) for the same base and height — a cone holds one-third of what a cylinder of the same footprint and height would hold. That relationship is worth memorizing as a sanity check. **Worked Example 1