Frustum Volume Calculator: Formula & Examples

This calculator works out the volume of a frustum — the portion of a cone or pyramid left after slicing off the top with a plane parallel to the base. You enter the bottom radius, top radius, and height (for a conical frustum), and it returns the volume in cubic units, plus the slant height and lateral surface area as useful extras.

Formula

The volume of a conical frustum is given by:

V = (π × h / 3) × (R² + R × r + r²)

Where:

• V = volume (cubic meters, cubic feet, etc.)
• h = perpendicular height between the two parallel faces (same unit as R and r)
• R = radius of the larger (bottom) circular face
• r = radius of the smaller (top) circular face

This is the same as the volume of a full cone minus the volume of the smaller cone that was removed. If you set r = 0, the formula reduces to the standard cone volume (π × R² × h / 3). If you set R = r, it becomes a cylinder (π × R² × h).

Worked Example

Using the default values from the calculator: bottom radius R = 2 m, top radius r = 1 m, height h = 3 m.

1. Square the bottom radius: 2² = 4
2. Multiply the two radii: 2 × 1 = 2
3. Square the top radius: 1² = 1
4. Add those three: 4 + 2 + 1 = 7
5. Multiply by π × h / 3: (π × 3 / 3) = π ≈ 3.14159
6. Volume = 3.14159 × 7 ≈ 21.99 m³

The calculator shows 21.99 m³, matching the hand calculation. The slant height is √((2−1)² + 3²) = √10 ≈ 3.16 m, and the lateral surface area is π × (2+1) × 3.16 ≈ 29.81 m².

What Is a Frustum and Where Is It Used?

A frustum is the 3D shape you get when you cut a cone or pyramid with a plane parallel to its base and discard the top part. The word comes from Latin, meaning “piece� or “broken off.� Everyday objects shaped like frustums include:

• Buckets and planters (usually tapered)
• Drinking cups and paper cups
• Lamp shades
• Concrete pipe transitions
• Certain machine parts and funnels

In construction, frustum volume calculations are needed for concrete in tapered footings, for soil in sloped excavations, and for material in hoppers or silos with tapered bottoms. In manufacturing, they help estimate material needed for molded or machined tapered parts.

How to Use the Calculator Correctly

Enter the two radii and the vertical height — not the slant height. The height must be measured perpendicular to the base, straight down the axis, not along the sloped side. If you only have the slant height (the distance along the side from bottom edge to top edge), you can find the vertical height with:

h = √(slant² − (R − r)²)

Then use that value in the calculator. Also ensure all three inputs are in the same unit. If you measure the radii in inches and the height in feet, the result will be wrong. Convert everything to one unit first.

Common Mistakes and How to Avoid Them

• Using diameter instead of radius: The formula requires the radius (half the diameter). If you measure across the top or bottom, divide by 2 first.
• Mixing up top and bottom: The formula is symmetric in R and r, so swapping them gives the same volume — but the slant height and surface area also stay the same, so it doesn’t matter for the volume.
• Using slant height as the height: This is the most common error. The height in the formula is the straight vertical distance, not the sloped side.
• Forgetting to cube the unit: If you enter meters, the volume is in cubic meters. If you need liters, multiply cubic meters by 1000. If you need gallons (US), multiply cubic meters by 264.172.
• Rounding too early: Keep intermediate values with more decimal places, then round the final answer. Rounding each step can change the result noticeably.

Limitations of the Formula

This formula applies only to a right circular frustum — one where the axis is perpendicular to the base and both faces are circles. If the shape is oblique (tilted), the volume formula changes and depends on the angle. For a frustum of a pyramid with a square or rectangular base, the volume is different: it uses the areas of the two bases, not radii. The general formula for any frustum (cone or pyramid) is:

V = (h / 3) × (A� + A₂ + √(A� × A₂))

where A� and A₂ are the areas of the two parallel faces. For a circular frustum, this reduces to the formula used here because A� = πR² and A₂ = πr².

The formula assumes the frustum is solid with straight sides. If the sides are curved (like a flower pot with a flare), the volume will be slightly larger or smaller, and a more complex integration may be needed.

Frequently Asked Questions

What is the difference between a frustum and a truncated cone?

They are the same thing. “Truncated cone� is a common synonym for a conical frustum. Both refer to the shape left when the top of a cone is cut off by a plane parallel to the base.

Can I use this calculator for a pyramid frustum?

No. This calculator is specifically for circular frustums. For a square or rectangular pyramid frustum, use the formula V = (h/3) × (A� + A₂ + √(A� × A₂)), where A� and A₂ are the areas of the two rectangular bases. For example, if the bottom base is 2 m × 3 m (area 6 m²), the top is 1 m × 1.5 m (area 1.5 m²), and the height is 2 m, the volume is (2/3) × (6 + 1.5 + √(6 × 1.5)) ≈ (2/3) × (7.5 + 3) = 7 m³.

How do I convert the volume to liters or gallons?

If your inputs are in meters, the volume is in cubic meters. Multiply by 1000 to get liters. For US gallons, multiply by 264.172. For imperial gallons, multiply by 219.969. If your inputs are in centimeters, the volume is in cubic centimeters; divide by 1000 for liters.

What if my frustum has different units for the radii and height?

Convert all measurements to the same unit before entering them. For example, if the radii are in inches and the height is in feet, convert the height to inches (multiply by 12) or convert the radii to feet (divide by 12). The volume will be in the cube of whatever unit you use.

Why does the calculator also show slant height and lateral surface area?

These are often needed alongside volume. Slant height is used for cutting sheet metal or fabric to wrap the frustum. Lateral surface area is used for painting, coating, or estimating material for the side. Having them in the same tool saves a second calculation.

Is the volume the same if I turn the frustum upside down?

Yes. The volume depends only on the two radii and the height, not on which face is on top. Swapping R and r gives the same result, so orientation does not matter.

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in r1 | Bottom radius | m | 2 | 0
in r2 | Top radius | m | 1 | 0
in h | Height | m | 3 | 0
out vol | Volume | m³ | (pi * h / 3) * (r1^2 + r1*r2 + r2^2) | 2
out slant | Slant height | m | sqrt((r1 - r2)^2 + h^2) | 2
out area | Lateral surface area | m² | pi * (r1 + r2) * sqrt((r1 - r2)^2 + h^2) | 2