Conical Frustum Volume Calculator
What Is a Conical Frustum?
A conical frustum is the portion of a right circular cone that remains after the tip has been sliced off by a plane parallel to the base. In other words, you start with a cone, cut it with a flat, horizontal slice, and keep the middle section between the two circular faces. The result is a solid with two different circular ends and straight, sloping sides.
This shape appears frequently in real life. Common examples include paper drinking cups, funnels, tapered buckets, lampshades, some flower pots, and decorative columns. In manufacturing and construction, you may encounter frustums in hoppers, silos, transition sections in ductwork, and many types of containers where a smooth taper is useful for strength, flow, or aesthetics.
Knowing the volume of a conical frustum helps you answer questions such as:
- How much liquid can a tapered cup, funnel, or tank hold?
- How much material (wood, metal, concrete, etc.) is needed or removed when shaping a tapered part?
- What is the capacity of a hopper used to store or dispense granular material?
This calculator lets you quickly compute that volume from three dimensions: the top radius, the bottom radius, and the vertical height between the two circular faces. All calculations are done directly in your browser; no data is sent to a server.
Conical Frustum Volume Formula
The volume of a conical frustum with:
- top radius r₁ (smaller circular end),
- bottom radius r₂ (larger circular end), and
- height h (perpendicular distance between the two circles)
is given by the standard formula:
V = (π × h × (r₁² + r₂² + r₁ × r₂)) / 3
In MathML form, the same equation can be written precisely as:
This expression combines three area-like terms inside the parentheses: one based on the top radius, one based on the bottom radius, and a cross term that accounts for the smooth taper between them. Because of this cross term, the volume is more than just the simple average of the volumes of two separate cones.
Relationship to the Cone Volume Formula
The familiar volume of a right circular cone with radius R and height H is:
Vcone = (π × R² × H) / 3
You can think of a frustum as the difference between two cones that share the same axis:
- a large cone with base radius r₂ and height H₂, and
- a smaller cone (the part that was cut off) with base radius r₁ and height H₁.
The frustum height is h = H₂ − H₁. Because the cutting plane is parallel to the base, the two cones are similar, so their dimensions are proportional. That similarity leads to the compact frustum formula above when you subtract the smaller cone’s volume from the larger cone’s volume.
How to Use This Calculator
- Measure the top radius r₁. This is half the diameter of the smaller circular end. You can measure the full diameter across the circle and divide by two.
- Measure the bottom radius r₂. This is half the diameter of the larger circular end.
- Measure the height h. Measure straight from the center of the top circle down to the center of the bottom circle, perpendicular to both faces. Do not measure along the sloping side.
- Enter the values in the calculator. Use any length unit you prefer (millimeters, centimeters, meters, inches, feet, etc.), but keep it consistent for all three inputs.
- Read the result. The calculator returns the volume in cubic units corresponding to the unit you chose (for example, if you enter centimeters, the result is in cubic centimeters, cm³).
Note that this tool expects non-negative values. If any dimension is zero, the resulting volume is zero. Negative dimensions do not make physical sense and should not be used.
Interpreting the Result
The output represents the internal volume of an ideal conical frustum that matches the dimensions you entered. In practical terms, you can interpret the value in several ways:
- Fluid capacity: If you are designing a container, hopper, or funnel, the result approximates how much liquid or granular material the shape can hold, assuming the interior matches the measured dimensions.
- Material usage: For woodworking, metalworking, or 3D printing, the volume gives you an estimate of how much material will be used in a solid tapered part, or how much material is removed when turning a frustum shape on a lathe.
- Weight estimates: Once you know the volume and the material density, you can estimate the mass: mass = density × volume. For example, concrete, steel, and plastic each have known densities you can multiply by the computed volume.
Always remember that the volume is in cubic units. If you mix units (for example, radii in centimeters and height in inches), the result will not correspond to a real physical quantity. Keep all length measurements in the same unit to ensure a meaningful cubic-unit output.
Worked Example
Suppose you have a tapered container shaped like a frustum with:
- top radius r₁ = 5 cm,
- bottom radius r₂ = 10 cm,
- height h = 20 cm.
Step 1: Compute the squared radii and cross term:
- r₁² = 5² = 25
- r₂² = 10² = 100
- r₁ × r₂ = 5 × 10 = 50
Step 2: Add these terms inside the parentheses:
r₁² + r₂² + r₁ × r₂ = 25 + 100 + 50 = 175
Step 3: Multiply by π and h, then divide by 3:
V = (π × h × (r₁² + r₂² + r₁ × r₂)) / 3
V = (π × 20 × 175) / 3
First, 20 × 175 = 3,500. So:
V = (π × 3,500) / 3 ≈ (3.14159 × 3,500) / 3 ≈ 10,995.6 / 3 ≈ 3,665.2 cm³
The frustum’s volume is approximately 3,665 cm³. If you want this in liters, recall that 1,000 cm³ = 1 liter, so the volume is about 3.67 liters.
Comparison Table: Frustum vs. Related Shapes
The frustum volume formula is related to those of cones and cylinders. The table below compares the key formulas and typical use cases.
| Shape | Dimensions | Volume Formula | Typical Uses |
|---|---|---|---|
| Right Circular Cylinder | radius r, height h | V = π × r² × h | Cans, pipes, uniform tanks, columns |
| Right Circular Cone | radius R, height H | V = (π × R² × H) / 3 | Funnels, conical piles, nozzles |
| Conical Frustum | top radius r₁, bottom radius r₂, height h | V = (π × h × (r₁² + r₂² + r₁ × r₂)) / 3 | Tapered cups, hoppers, transition ducts, decorative columns |
If your top radius and bottom radius happen to be equal (r₁ = r₂), the frustum becomes a cylinder, and the formula simplifies automatically to the cylinder volume formula. In that special case, the cross term preserves the correct volume without any extra work on your part.
Assumptions and Limitations
This calculator is based on an ideal geometric model. To use it correctly, keep the following assumptions and limitations in mind:
- Right circular frustum: The formula assumes the frustum is formed from a right circular cone. The central axis is perpendicular to both circular faces, and the cross-sections are perfect circles.
- Straight, linear sides: The sides between the two circular faces are straight (line segments), not curved or stepped. Shapes with bulging or irregular sides will not be modeled accurately by this formula.
- Parallel faces: The top and bottom circular faces are parallel to each other. If the cut is angled, the shape is an oblique frustum and requires more advanced methods to compute the exact volume.
- Consistent units: All three inputs must use the same length unit. Mixing units (for example, centimeters for one radius and inches for the height) will produce a numerically incorrect volume.
- Non-negative dimensions: Top radius, bottom radius, and height should be zero or positive. Negative dimensions are not physically meaningful.
- Geometric idealization: Real objects may have wall thickness, fillets, or manufacturing tolerances. This tool reports the ideal interior volume for the entered dimensions; adjust separately for wall thickness or fittings if needed.
- Input limits: Extremely large or extremely small values may exceed practical measurement accuracy or floating-point precision. For everyday engineering or construction uses, typical dimensions will work reliably.
By understanding these assumptions, you can better judge when this calculator is appropriate for your application and when a more detailed engineering model or measurement might be required.