Frustum Height Calculator – Solve for Height, Slant Height, or Volume
A frustum is the solid left over when a cone or pyramid has its tip sliced off by a plane parallel to its base. Cooling towers, buckets, lamp shades, hopper bins, and architectural spires are all frustum-shaped. When designing or measuring one of these objects, you rarely have every dimension at hand — often you know the two radii and either the slant height or the target volume, and need to find the perpendicular height instead. This calculator solves that problem in both directions.
Perpendicular Height vs. Slant Height
The perpendicular height (h) is the straight vertical distance between the two parallel bases. The slant height (l) is the distance measured along the sloped outer surface. These two lengths, together with the difference between the bottom radius R and top radius r, form a right triangle, so they are related by the Pythagorean theorem:
l = √(h² + (R − r)²)
Rearranged, this gives the height directly from a known slant height:
h = √(l² − (R − r)²)
Finding Height from Volume
If instead you know the desired volume of the frustum — for example, you are designing a hopper that must hold a specific quantity of material — the height can be back-calculated from the standard frustum volume formula:
V = (π × h / 3) × (R² + R×r + r²)
Solving for h gives:
h = 3V / (π × (R² + R×r + r²))
This mode is especially useful in tank and container design, where volume capacity is the starting requirement and the physical height is the unknown to be determined.
Full Frustum Analysis
Once the perpendicular height is known — whether entered directly or solved from slant height or volume — the calculator derives the complete geometry in one pass: slant height, volume, lateral surface area, and total surface area. The lateral surface area covers only the sloped side:
A_lateral = π × (R + r) × l
The total surface area adds the areas of both circular bases:
A_total = π × (R² + r² + (R + r) × l)
Geometric Validity
Not every combination of inputs describes a real frustum. Because h, l, and (R − r) form a right triangle, the slant height must always be at least as long as the difference between the two radii — that is, l ≥ |R − r|. If a slant height shorter than this is entered, the calculator flags the input as geometrically impossible rather than returning an imaginary result.
Real-World Applications
Structural and mechanical engineers use frustum height calculations when specifying cooling towers, silos, and transition ducts, where the vertical clearance is constrained but the capacity or lateral surface (for material estimation) is the design driver. Architects apply the same relationships to tapered columns and spires. In manufacturing, sheet-metal fabricators need the slant height to cut the correct pattern for a conical or pyramidal shell, then use the height to check that it fits the available vertical space.
Tips for Accurate Results
Keep all length inputs in the same unit — the calculator's unit selector applies uniformly to both radii and the slant height or height you enter. Double-check which radius is larger: R (bottom) must always be greater than r (top) for a standard frustum. When solving from volume, make sure the volume is expressed in the cube of your chosen length unit (for example, cm³ when your unit is centimetres) to avoid a mismatched result.