Conical Frustum Calculator

A conical frustum (also called a truncated cone) is the solid shape that remains when a smaller cone is sliced off the top of a larger cone by a plane parallel to its base. This ubiquitous geometric form appears in everyday objects such as drinking cups, lampshades, flower pots, and industrial components like reducer fittings and cooling towers. Understanding the exact measurements of a conical frustum is essential in engineering design, manufacturing, architecture, and material estimation.

The Conical Frustum Calculator computes the volume, slant height, lateral surface area, and total surface area from three measurements: the larger base radius $$R$$, the smaller base radius $$r$$, and the perpendicular height $$h$$. These calculations are fundamental in determining material requirements for sheet-metal fabrication, calculating liquid capacity of tapered vessels, and estimating concrete volumes for foundation piers.

The volume of a conical frustum is given by the formula:

$$V = \frac{\pi h}{3}\left(R^2 + Rr + r^2\right)$$

This elegant expression can be derived by subtracting the volume of the removed smaller cone from the original complete cone. It reduces to the standard cone volume formula $$V = \frac{1}{3}\pi R^2 h$$ when $$r = 0$$, and to a cylinder volume $$V = \pi R^2 h$$ when $$R = r$$.

The slant height $$l$$ is the distance measured along the lateral surface from the edge of one base to the edge of the other:

$$l = \sqrt{h^2 + (R - r)^2}$$

This follows directly from the Pythagorean theorem applied to the right triangle formed by the height, the radial difference, and the slant. The lateral surface area (the area of the curved side wall) uses this slant height:

$$A_{\text{lateral}} = \pi (R + r) l = \pi (R + r)\sqrt{h^2 + (R - r)^2}$$

The total surface area includes both circular bases:

$$A_{\text{total}} = A_{\text{lateral}} + \pi R^2 + \pi r^2$$

In practical applications, engineers use the lateral area to determine the amount of sheet metal or fabric needed to wrap a frustum-shaped structure. The volume calculation is critical when designing tapered tanks, hoppers, or funnels, where knowing the exact capacity ensures proper fluid storage or flow rates. Architects rely on frustum geometry when designing columns with entasis or tapered building elements. In civil engineering, frustum calculations are essential for determining the concrete volume in pile caps and tapered foundation footings. By entering your three key dimensions into this calculator, you obtain all essential measurements instantly, eliminating manual computation errors and saving valuable design time.