Where Frustums Appear

A conical frustum is the portion of a right circular cone that lies between two parallel planes cutting through it. Many everyday objects—from funnels and lampshades to drinking cups—use this tapering shape. Calculating its volume is useful in woodworking, manufacturing, and fluid measurement. While the formula might seem intimidating at first glance, it derives from the same principles as the familiar cone volume formula. This calculator lets you compute a frustum's volume from its top radius, bottom radius, and vertical height, with all math handled locally in your browser.

The word "frustum" comes from Latin and simply means a fragment or piece. In geometry, it refers to a solid obtained by slicing the top off a cone or pyramid. If the slice is parallel to the base, the resulting cross section is similar to the original base, leading to straightforward relationships between the radii and height. Because frustums appear in architectural columns, tapered storage bins, and cooking equipment, a fast volume calculation can aid in estimating materials and capacities.

Deriving the Volume Formula

The volume of a complete cone with base radius and height $H$ is:

To find the volume of a frustum, imagine two cones stacked tip to tip. The larger has base radius $\mathrm{r\_2}$ and height $\mathrm{H\_2}$, while the smaller "missing" cone has base radius $\mathrm{r\_1}$ and height $\mathrm{H\_1}$. The frustum height $h$ equals $\mathrm{H\_2}-\mathrm{H\_1}$. Because the slice is parallel to the base, similar triangles tell us:

Combining these relationships and subtracting the small cone volume from the large one yields the formula often written as:

Here $\mathrm{r\_1}$ represents the radius of the smaller end, $\mathrm{r\_2}$ the radius of the larger end, and $h$ the perpendicular distance between them. This equation works whether the frustum tapers down or flares outward, so long as the sides remain straight and symmetrical around a central axis.

Why Accurate Volume Matters

Volume estimates drive material purchasing, shipping calculations, and many other practical considerations. For example, when designing a funnel to transfer liquids, you need to know how much it can hold before overflow occurs. In woodworking, a decorative column might be shaped as a frustum, and determining its volume allows you to estimate the amount of wood removed on a lathe. Even in civil engineering, certain foundations use frustums to spread loads over a larger base, so understanding their capacity is essential.

The volume formula reveals that the frustum holds more than a simple average of two cones because of the additional $\mathrm{r\_1}\mathrm{r\_2}$ term. This cross product accounts for the gradual change in radius along the height. Failing to include it leads to underestimation, which could have consequences if you are pouring concrete or resin into a mold. By checking your design with this calculator, you gain confidence that your volume numbers are correct.

Units and Conversions

Because the formula multiplies radii by themselves, be sure all length measurements use the same units before performing the calculation. The result will then be in cubic units. If you enter radii in centimeters and height in centimeters, the volume comes out in cubic centimeters. To convert to liters, divide cubic centimeters by 1000. For cubic inches or gallons, additional conversion factors apply. A quick reference table is provided here:

Cubic Units	Convert to Liters
cm³	÷ 1000
m³	× 1000
in³	× 0.0164

Using the Calculator

Simply enter the radius of the smaller end $\mathrm{r\_1}$, the radius of the larger end $\mathrm{r\_2}$, and the vertical height $h$. Click the button and the script evaluates the formula above. The result appears in cubic units that match your input dimensions. All calculations occur locally using JavaScript, so you can experiment with different sizes instantly without sending data anywhere.

Remember that radii must be positive numbers. The ordering of $\mathrm{r\_1}$ and $\mathrm{r\_2}$ does not matter—if you swap them, you will obtain the same volume. In practice, it may be easier to measure diameters with calipers and then divide by two to obtain radii. Many manufacturing drawings specify diameters, so adjust accordingly.

Worked Example

Consider a truncated cone used as a hopper in a processing plant. The top opening radius is 15 cm, the bottom radius is 5 cm, and the vertical height is 30 cm. Plugging these into the formula gives:

This result translates to about 9.4 liters. Knowing the volume allows you to design the rest of the system—such as feed rates or storage capacity—around this value. You can verify by cross-checking with CAD software or manual calculations.

Exploring Variations

What happens if you double both radii while keeping the height the same? Because the radii terms are squared, the volume increases by a factor of four, all else being equal. On the other hand, doubling the height simply doubles the volume. Recognizing these proportional relationships helps when scaling designs or estimating how minor adjustments affect capacity. The frustum formula elegantly encapsulates these dependencies.

In some situations, you may need to find the missing height given a desired volume and radii. Although this calculator does not currently solve for height, rearranging the formula algebraically will yield a linear equation in $h$. Because there is only one instance of $h$ in the volume expression, you can isolate it on one side and solve directly. That technique is handy when designing a container to hold a specific volume.

Conclusion

The conical frustum appears in countless engineering and everyday contexts. By computing its volume with this simple formula, you ensure accurate material estimates and capacity measurements. This calculator performs the mathematics instantly in your browser, allowing you to adjust dimensions and immediately see the result. Whether you are crafting a custom flowerpot, designing a hopper, or just solving geometry homework, understanding the frustum's volume will give you deeper insight into the shape's properties.