Free Truncated Cone Volume Calculator

What Is a Truncated Cone (Frustum)?

A truncated cone, commonly referred to as a frustum, is a three‑dimensional solid formed by cutting off the top portion of a right circular cone with a plane parallel to its base. The resulting shape consists of two parallel circular faces: a smaller top face (radius $r$) and a larger bottom face (radius $R$), connected by a slanted lateral surface. The height $h$ is the perpendicular distance between these two circular faces. This geometric solid appears frequently in engineering, architecture, and everyday objects like buckets or lampshades.

Volume Formula for a Frustum

The volume of a truncated cone can be found by subtracting the volume of the small cone that was removed from the volume of the original complete cone. Using similarity ratios and algebraic simplification, we obtain the standard truncated cone formula:

$$V = \frac{1}{3} \pi h \left( r^2 + rR + R^2 \right)$$

where:

• $V$ is the volume,
• $h$ is the perpendicular height of the frustum,
• $r$ is the radius of the top circular face,
• $R$ is the radius of the bottom circular face.

This formula is the foundation of any frustum volume calculator and is essential for students and professionals alike.

How to Use This Geometry Calculator

The free online frustum volume calculator makes it easy to compute the volume without manual algebra:

1. Enter the known dimensions: Input the height $h$ and the radii $r$ and $R$ into the corresponding fields. Some calculators also accept the slant height $s$.
2. Choose your units: Select consistent units (e.g., cm, m, inches) so the result appears in the expected cubic unit.
3. Read the result: The calculator instantly applies the truncated cone formula and displays the volume.
4. Adjust as needed: Change any input value to see the updated volume, which is helpful for exploring different scenarios.

No formula memorization is required; the tool does the computation for you.

Worked Example

Let’s compute the volume of a frustum with the following dimensions:

• Height $h = 5\ \text{cm}$
• Top radius $r = 1\ \text{cm}$
• Bottom radius $R = 2\ \text{cm}$

Substituting into the formula:

$$\begin{aligned} V &= \frac{1}{3} \pi \times 5 \times (1^2 + 1 \times 2 + 2^2) \\ &= \frac{5\pi}{3} \times (1 + 2 + 4) \\ &= \frac{5\pi}{3} \times 7 \\ &\approx 36.65\ \text{cm}^3 \end{aligned}$$

This matches the result you would get instantly using the frustum volume calculator.

Important Considerations

• Always use the perpendicular height (the vertical distance between the two faces), not the slant height, unless the calculator specifically asks for the slant. If you have the slant height $s$ and the radii, you can compute the height using the Pythagorean relation: $h = \sqrt{s^2 - (R - r)^2}$.
• The formula works with any consistent unit system — the volume will be in the corresponding cubic units.
• For related volume computations (e.g., full cone, cylinder), you may also use a general cone volume calculator or a dedicated geometry calculator.