Truncated Cone (Frustum) Calculator

About the Truncated Cone (Frustum) Calculator

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.

When This Page Helps

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.

How to Use the Inputs

1. Enter the top radius (smaller circle) of the frustum.
2. Enter the bottom radius (larger circle) of the frustum.
3. Enter the height or slant height — the other is computed automatically.
4. Choose a preset to load common frustum dimensions.
5. Review volume, lateral area, and total surface area in the output cards.

Example Calculation

Tips & Best Practices

• If both radii are equal the frustum becomes a cylinder.
• If the top radius is zero the frustum is a complete cone.
• Slant height is always greater than the perpendicular height.
• For earthwork calculations, measure radii at the top and bottom of the cut.

A Cone with the Tip Removed

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.

Height Versus Slant Height

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.

Where Frustum Formulas Are Used

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.

Frequently Asked Questions

• What is a frustum?

  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.

• How do I find slant height from height?

  Slant height l = √(h² + (R − r)²), where h is the perpendicular height and R, r are the two radii.

• What if the top radius is larger than the bottom?

  The formula works the same — it is symmetric in R and r. Just enter the larger value as the bottom radius for conventional orientation.

• How is frustum volume derived?

  It can be derived by subtracting a smaller cone from a larger one, or by integrating circular cross-sections from bottom to top.

• What real objects are frustums?

  Buckets, flower pots, lampshades, cooling towers, paper cups, traffic cones (cut), and volcanic craters are all frustum-shaped.

• Is total surface area the same as lateral area?

  No. Total surface area includes the two circular end caps (πR² + πr²) in addition to the lateral (side) area.