Calculate volume, lateral surface area, and total surface area of a truncated cone (frustum). Enter top radius, bottom radius, and height or slant height.
The Truncated Cone Calculator computes every geometric property of a frustum — the solid formed by slicing a cone with a plane parallel to its base. Frustums appear everywhere: buckets, lampshades, drinking cups, cooling towers, volcano craters, and architectural columns. This calculator accepts the top radius (r), bottom radius (R), and either the perpendicular height (h) or the slant height (l). From these inputs it derives the volume using V = (πh/3)(R² + Rr + r²), the lateral (side) surface area using A_lat = π(R + r)·l, and the total surface area by adding the two circular end caps: A_total = π(R² + r² + (R + r)·l). The slant height is related to the other dimensions by l = √(h² + (R − r)²). Understanding frustum geometry is essential in civil engineering for calculating earthwork volumes, in manufacturing for designing tapered containers, and in mathematics for integral-based volume derivations. Use the preset buttons to explore common frustum shapes — from nearly cylindrical to steeply tapered. The formulas reference table summarizes every equation at a glance, and the visual dimension bars let you compare the top and bottom radii relative to the height.
Frustum calculations are common in real objects, but they mix circular geometry with a slanted side, so it is easy to confuse height, slant height, and the two radii. This calculator keeps those dimensions separate and computes the missing one automatically depending on the input mode. That makes it useful for container design, earthwork estimation, and manufacturing layouts where both volume and surface area matter and the shape is not a full cone or a simple cylinder.
Volume V = (πh/3)(R² + Rr + r²). Slant height l = √(h² + (R − r)²). Lateral area A_lat = π(R + r)·l. Total surface area A_total = πR² + πr² + π(R + r)·l.
Result: The slant height is about 8.246, the volume is about 410.50, and the total surface area is about 313.97.
For R = 5, r = 3, h = 8: l = √(64 + 4) = √68 ≈ 8.246. Volume = (π·8/3)(25 + 15 + 9) = (8π/3)·49 ≈ 410.50. Lateral area = π(8)(8.246) ≈ 207.18. Total area = 25π + 9π + 207.18 ≈ 313.97.
A truncated cone, or frustum, is easiest to understand as a full cone after the narrow top has been sliced away parallel to the base. That viewpoint explains why the volume formula includes all three radius terms $R^2 + Rr + r^2$ instead of looking like the cylinder formula. The solid still tapers, so both circular ends influence the result.
Many practical problems provide the slanted side because that is what you can measure directly on a physical object like a bucket or lampshade. Geometry problems, on the other hand, often provide the perpendicular height. Since those two measurements are related by a right triangle involving $R-r$, switching between them correctly is essential before using any area formulas.
Frustums appear in drainage structures, cooling towers, tapered cups, pots, and machined parts. Volume helps estimate capacity or material removal, while lateral surface area helps with coatings, labels, or fabrication. Reporting top and bottom cap areas separately is also useful when the ends are treated differently in a design or manufacturing workflow.
A frustum (or truncated cone) is the portion of a cone between two parallel planes cutting it. It has two circular faces of different radii.
Slant height l = √(h² + (R − r)²), where h is the perpendicular height and R, r are the two radii. Use this as a practical reminder before finalizing the result.
The formula works the same — it is symmetric in R and r. Just enter the larger value as the bottom radius for conventional orientation.
It can be derived by subtracting a smaller cone from a larger one, or by integrating circular cross-sections from bottom to top. Keep this note short and outcome-focused for reuse.
Buckets, flower pots, lampshades, cooling towers, paper cups, traffic cones (cut), and volcanic craters are all frustum-shaped. Apply this check where your workflow is most sensitive.
No. Total surface area includes the two circular end caps (πR² + πr²) in addition to the lateral (side) area.