Truncated Cone Calculator

Report an Issue

Help us improve by telling us what went wrong.

Embed this Tool

Customize and copy the embed code for your website.

Calculate Every Dimension of a Truncated Cone

A truncated cone, also called a frustum, is what you get when you slice the top off a cone parallel to its base. The result is a solid with two circular faces of different sizes connected by a sloped surface. Think of a lampshade, a flower pot, or a paper coffee cup. The Truncated Cone Calculator on ToolWard.com computes the volume, lateral surface area, total surface area, and slant height of any frustum from just three measurements.

The Formulas

Let R be the radius of the larger base, r be the radius of the smaller top, and h be the perpendicular height. The volume of a truncated cone is: V = (pi * h / 3) * (R squared + R * r + r squared). This elegant formula accounts for the tapering shape and reduces to the standard cone volume formula when r equals zero.

The slant height is: s = square root of (h squared + (R - r) squared). This is the distance along the sloped side from the edge of the larger base to the edge of the smaller top. The lateral surface area, which is just the sloped surface, is: pi * (R + r) * s. The total surface area adds both circular faces: pi * (R + r) * s + pi * R squared + pi * r squared.

Where Truncated Cones Show Up

Frustum geometry is everywhere in manufacturing and daily life. Disposable cups, buckets, traffic cones, lampshades, industrial hoppers, and concrete column footings are all truncated cones. Knowing the volume is essential for determining capacity, whether that means how much coffee a cup holds or how much grain a hopper stores. The lateral surface area tells you how much material is needed to fabricate the sloped wall, which drives cost estimates in sheet metal work, pottery, and textile manufacturing.

Civil engineers calculating the volume of earthwork for a sloped embankment or a tapered foundation footing use the frustum formula regularly. Calculating these volumes by hand is error-prone because the formula involves three terms that must be computed and summed before the final multiplication. The Truncated Cone Calculator eliminates arithmetic mistakes by performing all intermediate steps automatically.

How to Use the Calculator

Enter the radius of the larger base, the radius of the smaller top, and the perpendicular height. The calculator returns the volume, slant height, lateral surface area, and total surface area. If you know the diameters instead of the radii, simply divide by two before entering the values. All computation runs in your browser, so results appear instantly and no measurements leave your device.

Units must be consistent across all three inputs. If you enter radii in centimeters and height in centimeters, the volume will be in cubic centimeters and the areas in square centimeters. Mixing inches and centimeters will give a meaningless result.

Unrolling the Lateral Surface

One interesting property of a frustum is that its lateral surface, when unrolled and flattened, forms a shape called an annular sector, essentially a ring-shaped slice of a circle. This is how sheet metal workers create frustum shapes: they compute the unrolled dimensions, cut the flat pattern from a metal sheet, then roll and weld it into the three-dimensional shape. The lateral surface area calculation is central to this fabrication process, because it tells the worker how much sheet material is needed before cutting begins.

From Coffee Cups to Rocket Nozzles

The truncated cone is deceptively versatile. At one end of the complexity spectrum, it describes a simple paper cup. At the other end, it models the shape of rocket engine nozzles (converging-diverging nozzles have frustum sections), cooling tower profiles, and high-voltage electrical insulators. Regardless of the application, the math is the same, and the Truncated Cone Calculator handles it all reliably and instantly.