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Frustum Volume Calculator

Calculate the volume of a frustum (truncated cone) using V = (PI*h/3)*(R^2 + R*r + r^2), where R and r are the base radii and h is the height. Used in architecture, earthwork calculations, container design, and manufacturing of conical vessels.

Reviewed by Christopher FloiedUpdated

This free online frustum volume calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Radius of the larger circular base

Radius of the smaller circular base (0 for a full cone)

Perpendicular distance between the two bases

Results

Volume

1172.8613 cubic units

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Frustum Volume Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Frustum Volume Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Frustum Volume Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Frustum Volume Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Frustum Volume Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the volume of a frustum (truncated cone) using V = (PI*h/3)*(R^2 + R*r + r^2), where R and r are the base radii and h is the height. Used in architecture, earthwork calculations, container design, and manufacturing of conical vessels. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Frustum Volume Calculator

The Frustum Volume Calculator computes the volume of a frustum, which is the portion of a cone between two parallel planes cutting it. The formula V = (PI*h/3)*(R^2 + R*r + r^2) gives the exact volume when you know both base radii and the height. Frustums are everywhere in practical applications: drinking cups, lampshades, bucket shapes, architectural columns, earthwork embankments, cooling towers, and volcanic calderas. When the small radius r = 0, the formula reduces to the standard cone volume (PI*R^2*h/3). This calculator is essential for civil engineers computing earthwork volumes, manufacturers designing tapered containers, and architects working with truncated conical forms.

The Math Behind It

A frustum of a cone is obtained by cutting a cone with a plane parallel to its base. If the original cone has base radius R and height H, and the cut is made at height h from the base, the small radius at the cut is r = R*(H-h)/H. The frustum volume is the difference between the full cone and the removed tip. The full cone volume is (1/3)*pi*R^2*H. The removed tip has base radius r and height (H-h), so its volume is (1/3)*pi*r^2*(H-h). Using the relationship between R, r, H, and h, the frustum volume simplifies to V = (pi*h/3)*(R^2 + R*r + r^2). The expression R^2 + R*r + r^2 is reminiscent of the factoring identity a^3 - b^3 = (a-b)(a^2+ab+b^2). Indeed, the frustum volume equals (pi*h/3)*(R^3 - r^3)/(R-r) * (R-r) = pi*h*(R^3 - r^3)/(3*(R-r)) when derived from subtraction of cones. Historically, the frustum volume formula appears in ancient Egyptian mathematics. The Moscow Mathematical Papyrus (circa 1850 BCE) contains a problem solving for the volume of a frustum, making it one of the oldest known mathematical formulas. This predates Greek mathematics by over a thousand years. In modern engineering, the prismoidal formula generalizes frustum calculations to other shapes. The frustum formula is also used in numerical integration (Simpson's rule can be viewed as fitting parabolic frustum-like shapes). In civil engineering, earthwork volume calculations between cross-sections use frustum-based approximations.

Formula Reference

Frustum Volume

V = (pi*h/3) * (R^2 + R*r + r^2)

Variables: R = large base radius, r = small base radius, h = height

Worked Examples

Example 1: Tapered Bucket

A bucket has top radius 8 cm, bottom radius 4 cm, and height 10 cm.

Step 1:R = 8, r = 4, h = 10
Step 2:V = (pi*10/3) * (64 + 32 + 16) = (10.472) * 112 = 1172.86

The bucket volume is approximately 1172.86 cubic cm (about 1.17 liters).

Example 2: Cooling Tower Section

A cooling tower section has base radius 15 m, top radius 10 m, and height 20 m.

Step 1:V = (pi*20/3) * (225 + 150 + 100) = 20.944 * 475 = 9948.38

The section volume is approximately 9948.38 cubic meters.

Common Mistakes & Tips

  • !Forgetting the cross term R*r in the formula. The frustum volume is NOT (pi*h/3)*(R^2 + r^2). The middle term R*r is essential.
  • !Using the slant height instead of the perpendicular height. The height h must be measured perpendicular to the bases, not along the slant side.
  • !Confusing frustum formulas for surface area and volume. The lateral surface area uses pi*(R+r)*slant_height, which is a different formula entirely.

Related Concepts

Ellipsoid Volume

Another 3D shape with a formula involving pi and three dimensions, used for egg shapes and planetary bodies.

Torus Volume

Another non-standard 3D solid whose volume formula involves pi^2 rather than the frustum's simpler pi.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Ellipsoid Volume Calculator

Frequently Asked Questions

What happens when the small radius is zero?

When r = 0, the frustum becomes a complete cone. The formula gives V = (pi*h/3)*R^2, which is the standard cone volume formula. This confirms that the frustum formula generalizes the cone formula.

How old is the frustum volume formula?

The formula appears in the Moscow Mathematical Papyrus from ancient Egypt, dated to approximately 1850 BCE. This makes it one of the oldest known mathematical formulas, predating Greek mathematics by over a millennium.

How do I find the slant height of a frustum?

The slant height L = sqrt(h^2 + (R-r)^2), using the Pythagorean theorem on the right triangle formed by the height, the difference in radii, and the slant surface.