The frustum calculator above handles the solid you get when a cone or a pyramid has its top sliced off by a plane parallel to its base. Enter the two end dimensions and the perpendicular height and it returns the volume, the slant height, the lateral surface area, the total surface area, and the height of the complete cone or pyramid the frustum was cut from. It works for circular ends and for regular polygon ends with any number of sides.
Arb Digital publishes this page because the frustum is far more common in practice than the full cone, and its formulas are the ones people get wrong. Buckets, plant pots, lampshades, hoppers, drinking cups, roof sections, tapered chimneys, filter housings and machined bosses are all frusta. Our cone calculator covers full cones that run to a single point, and that is the boundary: if the shape has a flat top, it belongs here.
What This Frustum Calculator Does
It computes every standard quantity for a right frustum, meaning one whose two ends are parallel and whose axis is perpendicular to both. In cone mode you supply the two radii; in pyramid mode you supply the two side lengths and the number of sides, and the tool derives the polygon areas, the apothems and the face slant height from them.
It also reports the height of the original uncut solid, which is the quantity most often needed and least often given. Knowing it lets you check the volume a second way, and it tells you immediately whether the taper in front of you is consistent with the dimensions you measured.
Setting the top dimension to zero collapses the frustum back to a full cone or pyramid, and every formula degrades to the familiar one-third base times height. That is a useful check that the inputs are being read the way you intended.
How to Use It
- Choose the shape. Circular ends give a conical frustum; polygon ends give a pyramidal one, with four sides producing the common square-tapered case.
- Measure the perpendicular height. This is the vertical distance between the two faces. Measuring along the sloping side instead gives the slant height, which is longer and produces a volume that is too large.
- Enter both end dimensions. For a cone these are radii, not diameters. Halving a diameter is the single most common input error here and it changes the volume by a factor of four.
- Keep your units consistent. Areas come out in square units and volume in cubic units of whatever you entered.
- Read the surface split. The bars show how much of the total surface is the sloping side versus the two flat ends, which is what matters for material and coating estimates.
The Formulas and How They Are Calculated
For a conical frustum with radii R and r and height h, the volume is V = πh(R2 + Rr + r2)/3. Wolfram MathWorld's page on the conical frustum gives this form and notes that it generalises to any frustum by substituting the base areas. The slant height is l = √(h2 + (R − r)2), the lateral area is π(R + r)l, and the total surface area adds the two circles, πR2 + πr2.
The general form works for any frustum, whatever the cross-section: V = h(A1 + A2 + √(A1A2))/3, where A1 and A2 are the two end areas. MathWorld's page on the pyramidal frustum gives exactly this expression. For a regular n-sided polygon of side a the area is na2/(4 tan(π/n)), which is what the pyramid mode uses.
The slant height in pyramid mode is measured up the middle of a face, not along an edge, so it uses the difference of the two apothems rather than the two circumradii: l = √(h2 + (m1 − m2)2) with m = a/(2·tan(π/n)). Each lateral face is a trapezoid of area (a + b)l/2, so the lateral surface is n times that. The lateral edge is a longer, separate quantity built from the circumradii, and the tool reports it too because cutting stock to the edge length rather than the face slant is a classic fabrication error.
Why the Volume Formula Has That Middle Term
The natural guess for a tapered solid is to average the two end areas and multiply by the height. That guess is wrong, and the amount it is wrong by is worth understanding. The correct factor is the sum of the two areas plus their geometric mean, all divided by three — and the geometric mean of two numbers is always less than or equal to their arithmetic mean.
So averaging the ends always overestimates. With R = 5 and r = 3 the two circular areas are 78.54 and 28.27, whose average is 53.41; multiplied by a height of 4 that gives 213.6. The correct volume is 205.25, about four percent lower. The gap widens as the taper steepens: for a cone tapering all the way to a point, averaging the ends overestimates by fifty percent, since it gives half the base area times the height where the truth is a third.
The formula comes out of subtracting the small cone from the large one. The original solid has height H = hR/(R − r), and removing the tip of height H − h leaves precisely πh(R2 + Rr + r2)/3. That derivation is also the reason the middle term exists: it is what survives when you expand the difference of two cubes.
Slant Height Is Not the Height, and Not the Edge
Three different lengths run down the outside of a frustum and confusing them is the most frequent practical error. The perpendicular height h is the vertical gap between the two ends and is the only one that appears in the volume formula. The slant height l runs along the surface from the edge of one end to the edge of the other, and is the length that appears in every lateral-area formula. The lateral edge is the length of the sloping corner in a pyramidal frustum, and it is longer still.
For the default cone, h is 4 and l is √(16 + 4) = 4.472. Using 4.472 in place of 4 in the volume formula inflates the answer by about twelve percent. Using 4 in place of 4.472 in the lateral area formula understates the material needed by about eleven percent. Neither error announces itself, and both are easy to make when the only dimension you can physically reach with a tape measure is the sloping side.
If you have measured the slant and need the height, rearrange: h = √(l2 − (R − r)2). The measurement you cannot take is always recoverable from the two you can, and the same right-triangle relationship underlies our cone calculator and the surface work in our surface area calculator.
Flattening a Frustum for Fabrication
Making a conical frustum from sheet material means cutting an annular sector, not a rectangle or a trapezoid. The developed shape is the region between two concentric arcs. The outer arc has radius R·l/(R − r), the inner arc has radius r·l/(R − r), and both subtend the same angle, which in degrees is 360(R − r)/l.
The tool reports those three numbers under the result. For the default frustum they come out at an outer radius of 11.18, an inner radius of 6.708 and an included angle of 161.0 degrees. Cutting that sector and rolling it produces the exact side wall, with the arc lengths matching the two circumferences. A pyramidal frustum flattens far more simply, into n identical trapezoids each with parallel sides a and b separated by the face slant height, which is why sheet-metal transitions are often built as polygons even when the finished shape wants to be round.
A Worked Example You Can Check Yourself
The defaults are a conical frustum with R = 5, r = 3 and h = 4. The slant height is √(42 + (5 − 3)2) = √20 = 4.4721. The volume is π × 4 × (25 + 15 + 9)/3 = π × 4 × 49/3 = 196π/3 = 205.2507.
The lateral area is π(5 + 3) × 4.4721 = 112.3970. The two ends are 25π = 78.5398 and 9π = 28.2743, so the total surface area is 219.2112. The sloping side is therefore 51.3 percent of the total surface, which the bars confirm.
Now the cross-check. The original cone had height H = 4 × 5/(5 − 3) = 10, so its volume is π × 25 × 10/3 = 261.7994. The tip that was removed has height 6 and radius 3, giving π × 9 × 6/3 = 56.5487. Subtracting leaves 205.2507, which agrees with the direct formula to the last digit shown. That subtraction is the honest way to verify any frustum result, and it also settles arguments about whether a measured height was the perpendicular one. For the related solids, our cylinder volume calculator covers the untapered case, our pyramid volume calculator the full pointed pyramid, our sphere volume calculator the curved one, and our tank volume calculator the tapered vessels that combine several of them.
Arb Digital builds and maintains this whole free tool library, and every page is checked against a worked example before it goes live.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Entering diameters as radii — the volume depends on the square of each radius, so a factor of two in the input becomes a factor of four in the answer.
- Using the slant height as the height — the slant is always the longer of the two, and substituting it inflates the volume by a percentage that grows with the taper.
- Averaging the two end areas — the correct middle term is the geometric mean, not the arithmetic one, and averaging always overestimates.
- Adding both ends when one is open — a bucket or a hopper has no lid, so its material is the lateral area plus one end, not the total surface area.
- Cutting to the lateral edge instead of the face slant — on a pyramidal frustum the corner is longer than the middle of the face, and the two are not interchangeable.
Related Free Tools From Arb Digital
Handle the untruncated case with the cone calculator, the straight-sided version with the cylinder volume calculator, the full pointed solid with the pyramid volume calculator, curved surfaces with the sphere volume calculator, general shapes with the surface area calculator, or real vessels with the tank volume calculator. The free online tools hub lists every geometry tool we publish.
Frequently Asked Questions
It is the solid left when the top of a cone or pyramid is cut off by a plane parallel to its base. It has two parallel, similar faces of different sizes joined by a tapering side.
It is pi times the height times the quantity R squared plus R times r plus r squared, all divided by three, where R and r are the two radii. The middle term is what distinguishes it from a simple average of the ends.
Because the correct formula uses the geometric mean of the two areas as its middle term, and the geometric mean is never larger than the arithmetic mean. Averaging therefore always overestimates the volume, by fifty percent in the extreme case of a full cone.
The height is the perpendicular distance between the two flat ends, and it is what the volume formula needs. The slant height runs along the sloping surface between the two edges, and it is what the lateral surface area formula needs.
Multiply the frustum height by the bottom radius and divide by the difference of the two radii. The frustum volume is then the large cone minus the small cone that was removed, which is a useful independent check.
Yes. Switch to pyramid mode and enter the number of sides along with the bottom and top side lengths. The tool derives the polygon areas, the apothems and the face slant height from those.
The developed shape is a sector of an annulus. Its outer radius is R times the slant divided by the difference of the radii, its inner radius uses r in place of R, and the included angle in degrees is 360 times the radius difference divided by the slant height.
This page explains a geometric calculation for educational purposes only. It assumes an exact right frustum with parallel ends, and real components with wall thickness, draft angles or fillets need those allowances applied separately.
