Frustum Calculator — Truncated Cone
Calculate the volume, lateral surface area, and total surface area of a frustum.
A frustum is a truncated cone. R₁ = bottom radius, R₂ = top radius, h = height.
What Is the Frustum Calculator — Truncated Cone?
The Frustum Calculator computes the volume, lateral surface area, total surface area, and slant height of a frustum — a cone with the top cut off. Enter the bottom radius, top radius, and height to instantly find all geometric properties of this truncated cone shape.
Formula
How to Use
Enter the radius of the larger base (R), the radius of the smaller base (r), and the perpendicular height (h) of the frustum. The calculator returns volume, slant height, lateral surface area (the curved side), and total surface area (including both circular bases).
Example Calculation
Frustum with R=6 cm, r=3 cm, h=8 cm: Slant = √(64+9) = √73 ≈ 8.544 cm. Volume = π×8/3×(36+18+9) = π×8/3×63 ≈ 527.8 cm³. Lateral SA = π(6+3)×8.544 ≈ 241.5 cm². Total SA ≈ 241.5+π(36+9) ≈ 382.5 cm².
Understanding Frustum — Truncated Cone
A frustum of a cone is the solid obtained by cutting a cone with a plane parallel to its base, removing the smaller apex portion. The resulting shape has two circular faces (bases) of different radii and a curved lateral surface connecting them. Frustums are among the most practically important geometric solids in engineering and design.
The volume formula V = (πh/3)(R²+Rr+r²) can be understood as the average cross-sectional area (which varies quadratically between the two bases) multiplied by the height, with a geometric mean area term Rr accounting for the intermediate cross-sections. This is the prismatoid volume formula applied to conical geometry.
Frustum calculations arise in civil engineering (earthworks and excavation volume estimation), manufacturing (tapered containers and transitions), architecture (column design and decorative elements), and mathematics education. The same formulas apply to pyramidal frustums (e.g., a truncated square pyramid) by substituting the appropriate base areas.
Frequently Asked Questions
What is a frustum?
A frustum is the portion of a cone or pyramid between two parallel planes cutting the solid. A conical frustum has two circular bases of different radii connected by a curved lateral surface.
What is the difference between a frustum and a truncated cone?
They are the same shape. 'Frustum' is the mathematical term; 'truncated cone' is the descriptive term. Both refer to a cone with its apex removed by a cut parallel to the base.
Where are frustums found in real life?
Frustums appear as drinking cups (tapered cups), buckets, cooling towers, traffic cones, lamp shades, and geological formations. Many architectural columns have a frustum shape.
What happens when r=0?
When r=0, the frustum becomes a complete cone. The formulas reduce to V = πR²h/3 and lateral SA = πR×slant = πR√(h²+R²), the standard cone formulas.
Is this calculator free?
Yes, completely free with no account required.
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