Frustum Cone Volume Calculator

Frustum (truncated cone) volume

Take a cone and slice off the tip with a plane parallel to the base. What's left is a frustum. Its volume comes out to V = (π·h/3)·(R² + r² + R·r), where R is the larger radius (base), r the smaller radius (top), and h the height between the two circular faces. Example: a conical cup with R = 4 cm, r = 2 cm, h = 10 cm gives V = (π·10/3)·(16 + 4 + 8) ≈ 293 cm³ ≈ 293 mL. Set r = 0 and you're back to a full cone (V = πR²h/3); set R = r and it's a cylinder (V = πR²h).

Applications

It shows up in packaging design (conical cups, yogurt pots, flowerpots), in waste-bin sizing under ABNT NBR 13230, in industrial nozzles and reducers, and in the cooling towers of power plants (where the hyperboloid is the limit case). You'll also meet it in lampshades and decorative objects, and in civil engineering for reservoirs and silos with conical hoppers.

FAQ

Why three terms (R² + r² + R·r)? When you integrate the area of a circle whose radius grows linearly with height from R to r, you don't get the plain average of R² and r². You get a weighted blend of three "intermediate" circles, and that's where the extra R·r term comes from.

What happens when r = 0? The top shrinks to a point, the frustum becomes a complete cone, and the formula collapses to V = πR²h/3.

What is the slant height (generatrix)? g = √(h² + (R−r)²). You need it to find the lateral area, but it plays no part in the volume.