Conical Frustum Volume Calculator

Introduction

A conical frustum is one of those shapes that shows up everywhere once you know its name. It is the solid you get when the tip of a cone is cut off parallel to the base, leaving a smaller circular face on one end and a larger circular face on the other. If you have ever measured a paper cup, a hopper, a funnel section, a flower pot, a lampshade frame, or a tapered duct transition, you have handled a frustum. This calculator is designed to answer the practical question that usually follows: how much space is inside that tapered shape?

This page lets you compute volume from three measurements: the radius of one circular end, the radius of the other circular end, and the vertical height between them. Enter all three in the same unit and the calculator returns the volume in matching cubic units. In other words, if your inputs are in centimeters, the output is in cubic centimeters; if your inputs are in inches, the result is in cubic inches. Because the formula is symmetric in the two radii, the calculator still works even if your object is turned upside down and the top radius is larger than the bottom radius.

That makes the tool useful in many settings. A builder might estimate the concrete in a tapered footing. A woodworker might calculate how much stock is removed when turning a tapered form on a lathe. A process engineer might estimate the capacity of a transition hopper. A product designer might compare the internal volume of several cup sizes before prototyping. The math is the same in every case, and this calculator packages it into a quick browser-based check.

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 plain language, you begin with a cone, remove the pointed top with a flat cut, and keep the section between the two circular faces. The result is a solid with straight sloping sides and two different circular ends.

This shape appears frequently in real objects because a gradual taper is often more useful than a sudden change in size. Tapered walls can make pouring easier, improve stackability, reduce material use, and create a cleaner transition between two openings. That is why frustums show up in containers, architectural features, nozzles, fittings, and parts meant to guide fluids or bulk material.

When you know the volume of a frustum, you can answer questions such as how much liquid a tapered cup can hold, how much internal space a hopper provides, or how much material is contained in a solid decorative turning. The same calculation can also support weight estimates after you multiply volume by density.

Conical Frustum Volume Formula

The standard volume formula for a conical frustum uses three measurements:

• r₁, the radius of one circular end,
• r₂, the radius of the other circular end, and
• h, the perpendicular height between those two circles.

The volume is:

V = (π × h × (r₁² + r₂² + r₁ × r₂)) / 3

The same relationship is shown below in MathML so the formula keeps its mathematical structure:

There are two important ideas hidden in that compact expression. First, volume scales directly with height. Double the height while keeping the radii fixed and the volume doubles. Second, the radii matter more dramatically because the formula includes squared terms as well as the cross term r₁r₂. That is why even modest changes in width can noticeably change capacity.

The formula is also symmetric in r₁ and r₂. Swapping the two radii does not change the result. So although the inputs are labeled top and bottom for convenience, the mathematics does not care which end is physically higher.

How the Frustum Formula Connects to Cone Volume

The familiar volume of a right circular cone with radius R and height H is V = (π × R² × H) / 3. A frustum can be understood as a larger cone with a smaller similar cone removed from its tip. Because the cut is parallel to the base, the two cones are similar, which means corresponding lengths stay in proportion. Subtracting the smaller cone volume from the larger cone volume leads to the frustum formula above.

This point of view is useful because it explains why the expression is not simply the average of a small cone and a large cone. The taper contributes continuously along the full height, and that contribution is captured by the squared-radius terms and the cross term.

How to Use This Calculator

The calculator is straightforward, but accuracy depends on measuring the right dimensions.

1. Measure the top radius. If you know the diameter instead, divide it by two. Measure from the center of the circle to its edge, not across the full opening.
2. Measure the bottom radius. Again, you can measure the diameter and divide by two if that is easier in the field.
3. Measure the height. Use the straight vertical distance between the two circular faces. Do not use the sloping side length.
4. Keep the units consistent. Use one length unit for every entry. Mixing centimeters and inches will produce a meaningless numeric result.
5. Submit the form. The result will appear immediately below the button in cubic units.

For this specific calculator, the input validation requires positive numbers. A perfect geometric frustum with a zero dimension would collapse into a degenerate edge case, so the form intentionally rejects zero and negative values and asks for valid dimensions instead.

If you are measuring a physical object, make sure you are using the inside dimensions if you want internal capacity. If you use outside dimensions on a cup, tank, or hopper, you will slightly overestimate the usable volume because wall thickness is ignored.

Interpreting the Result

The output is the volume of the idealized frustum defined by your three measurements. In practical terms, that can represent internal capacity, total solid material, or removed material depending on the job you are doing.

For example, if you are working with a container, the result tells you how much it can hold in the same cubic unit system implied by the inputs. If the inputs are in centimeters, the answer is in cubic centimeters, and you can convert to liters by dividing by 1,000. If you are modeling a solid part, the same number can be multiplied by density to estimate mass or shipping weight.

It is also helpful to compare the result to nearby shapes. If the two radii are equal, the frustum becomes a cylinder and the formula reduces naturally to V = πr²h. If one radius becomes very small, the shape approaches a full cone. These limiting cases are a good mental check that your dimensions make sense.

Worked Example

Suppose a tapered container has a top radius of 5 cm, a bottom radius of 10 cm, and a vertical height of 20 cm.

Start by computing the three terms inside the parentheses:

• r₁² = 5² = 25
• r₂² = 10² = 100
• r₁ × r₂ = 5 × 10 = 50

Now add them together:

25 + 100 + 50 = 175

Next substitute into the full formula:

V = (π × 20 × 175) / 3

Multiply height by the parenthetical total first:

20 × 175 = 3,500

So the volume becomes:

V = (π × 3,500) / 3 ≈ 3,665.2 cm³

That means the container holds about 3.67 liters, because 1,000 cubic centimeters equals 1 liter. The example shows how quickly capacity grows when both the height and the larger radius are substantial.

Comparison with Related Shapes

Frustums sit between cylinders and cones in a very natural way. The table below is helpful when you are deciding whether you really have a frustum or a simpler special case.

If r₁ = r₂, the frustum formula collapses to the cylinder formula because r₁² + r₂² + r₁r₂ becomes 3r², and the factor of 3 cancels with the denominator. That is a nice built-in check of the equation.

Assumptions and Limitations

Every geometry calculator depends on a model, and this one is no exception. The result is exact for the model but only approximate for real objects that depart from it.

• Right circular geometry: The formula assumes both ends are circles and share a common central axis.
• Parallel circular faces: The two ends must be parallel. If the cut is angled, the solid is an oblique frustum and needs a different treatment.
• Straight side walls: The taper must be linear. Bulging, curved, ribbed, or stepped sides are not represented accurately.
• Consistent units: All three inputs must use the same length unit so the final answer has coherent cubic units.
• Ideal dimensions: Real parts may include wall thickness, rounded corners, seams, coatings, or manufacturing tolerances that slightly change true capacity.

In practice, these assumptions are reasonable for many engineering, construction, and shop-floor estimates. When precision matters, treat the result as the geometric baseline and then adjust for wall thickness, fittings, or measured irregularities.

Why This Calculator Is Useful

What makes the frustum shape tricky is that it looks simple, but estimating its volume by eye is rarely reliable. A slight widening at one end can add more volume than you expect because the radii do not act linearly in the equation. This calculator removes that guesswork. It gives you a fast answer, supports any consistent unit system, and keeps the math local in your browser so you can experiment with dimensions freely.

If you are comparing designs, try changing one measurement at a time and observing the result. Increasing height changes the answer in a steady, proportional way. Increasing either radius tends to have a stronger effect, especially when both ends are already wide. That intuition becomes even clearer in the mini-game below, where you tune frustums by feel and see how quickly the volume gauge responds.