Why can I not just average the two diameters?

Because volume depends on the SQUARE of the radius, and squaring does not survive averaging. The square of an average is always less than the average of the squares whenever the two numbers differ — that is a mathematical fact, not a quirk of cones — so averaging the diameters throws away exactly the part that makes the wide end wide. The shortcut is therefore wrong for every taper, and always wrong in the same direction: it UNDER-states the volume. How much depends on how severe the taper is. On a gentle taper the error is a per cent or two and nobody notices. On a hopper going from two metres down to four hundred millimetres it is about an eighth, which on a concrete pour is a wasted delivery or a short one. The page shows both figures side by side so the size of the error is visible rather than asserted.

What is the difference between vertical height and slant height?

The vertical height is the straight up-and-down distance between the two circular faces; the slant height is the distance measured along the sloping side, from the edge of one face to the edge of the other. The slant is always the longer of the two, and the steeper the taper the bigger the gap. This matters because the slant is often the one that is easy to measure — you can run a tape up the outside of a hopper or a silo cone, whereas the vertical height needs a level and a plumb line. Entering a slant where a vertical height is wanted inflates the answer, and on a sharply tapered cone it can inflate it badly. If the slant is all you have, the vertical height is the square root of the slant squared minus the difference of the two radii squared — the two of them and the taper form a right triangle.

Is this the right shape for a stockpile?

For a levelled one, roughly. A pile that has been tipped and then had its top scraped flat is a frustum, and a pile simply tipped and left is a cone, which this page gives if you enter zero for the smaller diameter. But treat either as an estimate with a wide band rather than a measurement. Material finds its own angle of repose, which varies with the material, its moisture and how it was placed; the base is rarely a true circle, especially where a loader has been working one side; and the ground underneath is rarely flat, so the apparent height from outside can include a hollow or sit on a hump. There is also the bulking question, which the geometry cannot answer: the same material occupies noticeably more space loose in a pile than it did in the ground or will once it is compacted in place, and which of those three numbers you actually want depends on whether you are buying it, hauling it or placing it.

Does a hopper really hold what the geometry says?

No, and the difference is the reason hopper design is its own discipline. The geometric volume is an upper bound that assumes the material behaves like a liquid, and granular materials do not. They arch: particles wedge against each other across the converging walls and form a stable bridge that leaves a void beneath it. They rathole: material flows only down a central channel while the rest stands still against the wall. They also leave dead zones in the corner between the cone and the cylinder above it, where material sits until somebody hits the side with a hammer. The working capacity is therefore always less than the shape holds, and the gap depends on the material's flow properties and the wall angle rather than on anything in this arithmetic. Use the figure here for how much fits, and expect the amount that actually comes out to be less.

Materials & Quantities

Cone Frustum Volume Calculator — Hoppers and Tapered Piers

Volume of a truncated cone from its two diameters and its height, and how much the common mean-diameter shortcut under-states it.

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Every formula citedNamed published standards
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Market

Imperial · sales tax

Larger diameter

The diameter of the wider of the two parallel faces.

For a hopper this is the top opening; for a levelled stockpile it is the base on the ground; for a tapered pier it is usually the bell at the bottom. Which end is which does not change the answer — the expression is symmetric in the two radii — so enter them in whichever order is natural.

Smaller diameter

The diameter of the narrower face. Enter zero for a full cone.

Zero is a legitimate value and gives a complete cone, which is the right shape for a tipped pile that has not been levelled. A hopper's outlet diameter goes here, measured at the throat rather than at the valve below it.

Vertical height

The straight vertical distance between the two faces.

VERTICAL, not the slant length up the sloping side. The slant is the longer of the two and the difference grows with the taper, so measuring along the sheet metal of a hopper and entering that here overstates the volume. If you only have the slant, take the vertical as √(slant² − ((R − r))²).

Frustum volume

68.98 ft³

High confidence

Diameters taper 1.98 to 0.41, a ratio of 0.21. Averaging the two diameters and treating this as a cylinder gives a figure 12.7% SHORT — the mean-diameter shortcut under-states every frustum, because volume goes with the square of the radius and the square of an average is less than the average of the squares.

As a liquid capacity: 515.98 gal
Mean-diameter shortcut, for comparison: 60.24 ft³
What that shortcut misses: 8.74 ft³
Shortcut error: 12.66 %

Then change the inputs to see how far the answer moves.

Show calculation logic

How this was calculated

Formula source(s)

The volume of a frustum of a right circular cone is (π h / 3)(R² + R·r + r²), exact for any two radii and vertical height. Setting r = R returns a cylinder and setting r = 0 returns a cone, which is the check that the expression is the general case rather than an approximation
It is the prismoidal rule in disguise: R² + R·r + r² is (A₁ + √(A₁A₂) + A₂)/π, so the mid-section is the GEOMETRIC mean of the two ends rather than the arithmetic mean. The same expression appears in earthworks volumes between cross-sections, for the same reason
The height wanted is the VERTICAL height between the two parallel faces, not the slant height along the sloping side. Slant height is longer and using it inflates every result

Inputs used

Larger diameter: 6.5 ft
Smaller diameter: 1.33 ft
Vertical height: 5 ft

Intermediate steps

As a liquid capacity: 515.98 gal
Mean-diameter shortcut, for comparison: 60.24 ft³
What that shortcut misses: 8.74 ft³
Shortcut error: 12.66 %

Final result68.98 ft³

Confidence note: Diameters taper 1.98 to 0.41, a ratio of 0.21. Averaging the two diameters and treating this as a cylinder gives a figure 12.7% SHORT — the mean-diameter shortcut under-states every frustum, because volume goes with the square of the radius and the square of an average is less than the average of the squares.

What this calculation does not cover

This is a RIGHT circular frustum — both faces circular, parallel, and centred on the same axis. A hopper whose outlet sits off to one side, or whose sides are flat plates rather than a rolled cone, is a different solid and this over- or under-states it depending on the offset.
The height must be the vertical one. Slant height measured up the sloping face is always longer, and on a steep taper using it in place of the vertical height inflates the answer substantially.
A stockpile is not a frustum until it has been levelled, and even then only roughly. Tipped material finds its own angle of repose, the base is rarely a circle, and the ground underneath is rarely flat — treat a pile figure as an estimate with a wide band rather than a measurement.
Nothing here allows for the material's bulking. A cubic metre of excavated soil or crushed stone occupies more space loose than it did in the ground or will after compaction, and which of those three volumes you want is a separate question with its own page.
For a hopper, the volume computed is the geometric capacity, not the working capacity. Material bridges above the outlet, leaves a rathole down the middle, or sits in a dead zone at the wall, and the usable amount is always less than the shape holds.

Estimated cost — your price

Price per ft³ ($)

This site holds no price list for this material — local prices vary too much to publish honestly. Enter your supplier's price and the result is costed with it.

Schematic, drawn to the proportions you entered — not to scale on screen.

Computed in your browser — nothing you enter is uploaded. Presented in US customary units and US trade terminology. Where a formula follows a published standard, that standard and its edition are cited beside it on this page; where none governs, the page says so. Local amendments override model codes — verify against the code in force where you build.

Sources checked 2026-09-15 · v1.0.0

Regulatory standards & verification citations3

The volume of a frustum of a right circular cone is (π h / 3)(R² + R·r + r²), exact for any two radii and vertical height. Setting r = R returns a cylinder and setting r = 0 returns a cone, which is the check that the expression is the general case rather than an approximation
It is the prismoidal rule in disguise: R² + R·r + r² is (A₁ + √(A₁A₂) + A₂)/π, so the mid-section is the GEOMETRIC mean of the two ends rather than the arithmetic mean. The same expression appears in earthworks volumes between cross-sections, for the same reason
The height wanted is the VERTICAL height between the two parallel faces, not the slant height along the sloping side. Slant height is longer and using it inflates every result

Cite this page

Your workspace

Most jobs need more than one number. Add the calculators you need next and they open right here, underneath this one — your figures stay on screen and nothing is lost to a page change.

How to calculate cone frustum volume — hoppers and tapered piers in 4 steps

Larger diameterThe diameter of the wider of the two parallel faces.
Smaller diameterThe diameter of the narrower face. Enter zero for a full cone.
Vertical heightThe straight vertical distance between the two faces.
Frustum volumeThe tool computes the frustum volume from those figures and shows the formula, its sources, and a confidence rating alongside it.

Frustum volume by larger diameter

Page defaults, not your figures above.

Larger diameter	Frustum volume (ft³)
4 ft	29.6
6 ft	58.7
8 ft	98.2
10 ft	148
12 ft	208

Frequently asked questions

Why can I not just average the two diameters?: Because volume depends on the SQUARE of the radius, and squaring does not survive averaging. The square of an average is always less than the average of the squares whenever the two numbers differ — that is a mathematical fact, not a quirk of cones — so averaging the diameters throws away exactly the part that makes the wide end wide. The shortcut is therefore wrong for every taper, and always wrong in the same direction: it UNDER-states the volume. How much depends on how severe the taper is. On a gentle taper the error is a per cent or two and nobody notices. On a hopper going from two metres down to four hundred millimetres it is about an eighth, which on a concrete pour is a wasted delivery or a short one. The page shows both figures side by side so the size of the error is visible rather than asserted.
What is the difference between vertical height and slant height?: The vertical height is the straight up-and-down distance between the two circular faces; the slant height is the distance measured along the sloping side, from the edge of one face to the edge of the other. The slant is always the longer of the two, and the steeper the taper the bigger the gap. This matters because the slant is often the one that is easy to measure — you can run a tape up the outside of a hopper or a silo cone, whereas the vertical height needs a level and a plumb line. Entering a slant where a vertical height is wanted inflates the answer, and on a sharply tapered cone it can inflate it badly. If the slant is all you have, the vertical height is the square root of the slant squared minus the difference of the two radii squared — the two of them and the taper form a right triangle.
Is this the right shape for a stockpile?: For a levelled one, roughly. A pile that has been tipped and then had its top scraped flat is a frustum, and a pile simply tipped and left is a cone, which this page gives if you enter zero for the smaller diameter. But treat either as an estimate with a wide band rather than a measurement. Material finds its own angle of repose, which varies with the material, its moisture and how it was placed; the base is rarely a true circle, especially where a loader has been working one side; and the ground underneath is rarely flat, so the apparent height from outside can include a hollow or sit on a hump. There is also the bulking question, which the geometry cannot answer: the same material occupies noticeably more space loose in a pile than it did in the ground or will once it is compacted in place, and which of those three numbers you actually want depends on whether you are buying it, hauling it or placing it.
Does a hopper really hold what the geometry says?: No, and the difference is the reason hopper design is its own discipline. The geometric volume is an upper bound that assumes the material behaves like a liquid, and granular materials do not. They arch: particles wedge against each other across the converging walls and form a stable bridge that leaves a void beneath it. They rathole: material flows only down a central channel while the rest stands still against the wall. They also leave dead zones in the corner between the cone and the cylinder above it, where material sits until somebody hits the side with a hammer. The working capacity is therefore always less than the shape holds, and the gap depends on the material's flow properties and the wall angle rather than on anything in this arithmetic. Use the figure here for how much fits, and expect the amount that actually comes out to be less.

Preliminary estimate, not certified engineering. This tool produces an indicative quantity calculation for planning purposes only — it is not a certified structural analysis, a guaranteed material takeoff, or a substitute for building department approval. Always verify measurements on-site and have a licensed contractor or structural engineer review any load-bearing, code-sensitive, or safety-critical work before purchasing materials or starting construction. Spotted an arithmetic or standards error? Report it to contact@craftquantities.com with your inputs — a confirmed fix gets a permanent check of its own, so the same mistake cannot come back.