Truncated Cone Volume

Frustum = truncated cone.

How the Math Works

The truncated cone volume formula V = πh/3(R²+Rr+r²) calculates the space inside a cone with its top cut off by a plane parallel to the base. Here, R is the radius of the larger base, r is the radius of the smaller top base, h is the vertical height between them, and π (pi) is the ratio of a circle's circumference to its diameter. The formula cleverly combines the areas of both circular bases (R² and r²) plus their geometric mean (Rr), then multiplies by height and divides by 3 - essentially averaging the two base areas in a way that accounts for the tapering shape, similar to how a full cone's volume uses one base area multiplied by height and divided by 3.

Practical Applications

To use this calculator practically, measure the radius of the larger circular end (R) and the smaller circular top (r) of your truncated cone-shaped object, then determine the perpendicular height (h) between these two bases. For example, if designing a lampshade with a bottom diameter of 14 inches (R=7), top diameter of 6 inches (r=3), and height of 12 inches, plug these values into the formula to find the material volume needed. This calculation is essential for manufacturing, construction projects involving tapered structures, or determining liquid capacity in containers with varying diameters.

Day-to-Day Use

This calculation appears in everyday scenarios like determining how much coffee your tapered mug holds, calculating concrete volume for a truncated concrete column, or figuring out how much grain fits in a conical pile with the top removed. It's useful when working with lampshades, buckets, buckets, funnels, or any container that widens or narrows from top to bottom. Understanding this also helps estimate material requirements for DIY projects involving cut-off cones, making it a practical tool for homeowners, crafters, and anyone working with tapered cylindrical objects.

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