Frustum Calculator

A frustum is what is left of a cone after the tip is sliced off with a cut parallel to the base, which is why it is also called a truncated cone. Its volume is V = (π h ÷ 3) × (R² + R r + r²), where R and r are the radii of the two circular faces and h is the straight vertical distance between them. This frustum calculator returns that volume together with the slant height, both surface areas (lateral and total), the capacity in litres and in both kinds of gallon, and the flat pattern you would cut to build one.

Almost every tapered round container in the house is a frustum. Buckets and pails, plant pots, lampshades, drinking glasses, tapered cake tins and the outlet section of a hopper or a silo all share the shape, and all of them are measured the same way: the width across the bottom, the width across the top, and the height between the two faces. Because most people measure a bucket with a tape laid across the rim rather than from the centre, the calculator takes full widths by default and halves them for you.

Where the formula comes from

The honest way to build a frustum is to start with the whole cone and take the tip away. Call the height of the removed tip a. The triangle formed by the cut and the triangle formed by the full cone are similar, so their sides sit in the same ratio: a ÷ r = (a + h) ÷ R. Rearranging that gives a = h r ÷ (R − r), so the complete cone stands H = h R ÷ (R − r) tall. Those two results are worth having on their own, because they tell you exactly how far above the small end the missing point would have sat.

Now subtract. The whole cone holds one third of π R² H and the removed tip holds one third of π r² a. Substitute the two heights, put everything over the common denominator R − r, and the numerator becomes the difference of two cubes, R³ − r³. That factors into (R − r)(R² + R r + r²), the R − r cancels, and what survives is (π h ÷ 3)(R² + R r + r²). The one third factor is inherited from the cone, which the cone volume calculator covers in full.

The middle term is where intuition fails. A frustum is not the average of the two end circles multiplied by the height. Averaging πR² and πr² would give a figure that is too large, because the cross section shrinks steadily from bottom to top rather than jumping halfway. The R r term is the geometric middle between the two circles, and it is what drags the answer down to the correct value.

Slant height, and why the volume ignores it

Look at the frustum in cross section and the sloped edge is the hypotenuse of a right triangle whose vertical leg is h and whose horizontal leg is the difference between the radii. Pythagoras gives s = √(h² + (R − r)²). A frustum with radii of 4 and 1 and a height of 4 has a slant of exactly 5, which is the familiar 3, 4, 5 triangle hiding in the taper.

Volume never uses s, because volume only cares about how much cross section sits at each level and how far apart the levels are. Surface area is the opposite: the sloped face has to be measured along its own surface, so every area formula on this page uses s and none of them use h. That single distinction explains most of the wrong answers people get with a truncated cone.

Lateral and total surface area

The lateral area is the sloped band on its own, L = π (R + r) s. Read it as the average circumference of the two rims multiplied by the slant, which is exactly what it is once the band is unrolled. That is the number you want for the fabric on a lampshade, the sheet metal in a hopper wall, the label wrapped round a tapered cup, or the paint on the outside of a pail.

Total area adds the two flat faces: A = π (R² + r² + (R + r) s). Most real containers are open at one end, so take one face back off when you are buying material. A lampshade has no faces at all, a bucket has one, and a sealed hopper section has two.

Measuring a real container

Geometry is the easy half. Getting the three numbers off a physical object is where the errors live, and there are two decisions to make before the tape comes out.

The first is inside or outside. Capacity is an inside measurement. Walls on a plastic pail run a few millimetres thick, and on a terracotta pot or a cast tin they can be far more, so outside diameters inflate the answer. The second is brim or fill line. A standard United States five gallon pail measures about 11.91 in across the top, 10.33 in across the base and stands 14.5 in tall. Run those outside figures to the brim and this calculator returns about 1,411 cubic inches, which is 6.11 US gallons. The gap between that and the label on the side is the wall thickness, the rim and the headroom the lid needs. Neither number is wrong. They answer different questions, and a capacity figure is only meaningful once you say which one you asked.

Litres, US gallons and Imperial gallons

Cubic centimetres and cubic inches are awkward to picture, so the calculator converts every result into litres and into both gallons. A US gallon is defined as 231 cubic inches, which is 3.785411784 litres. An Imperial gallon is 4.54609 litres, near 277.42 cubic inches. The Imperial gallon is about 20 percent larger, so the same pot that holds 6.11 US gallons holds only 5.08 Imperial gallons. That is why this page never prints a bare figure labelled gallons. Both are always shown, always named, and the reader picks.

Cutting a frustum from flat sheet

Unroll the sloped band and it does not become a rectangle. It becomes an annular sector, the region between two arcs that share a centre. Extend the shape to its missing apex first and the pattern follows directly: the outer radius is s R ÷ (R − r), the inner radius is s r ÷ (R − r), and the included angle is θ = 360° × (R − r) ÷ s. Draw both arcs from one centre point, cut along the two straight edges, and roll.

There is a self check that costs nothing and catches a mis-measured taper immediately. The outer arc has to wrap the big end, so its length must equal 2πR, and the inner arc has to wrap the small end, so its length must equal 2πr. If either arc comes up short on the sheet, the numbers that produced it were wrong.

Worked examples

1. 1Plant pot

   25 cm across the rim, 18 cm across the base, 22 cm deep. V = (π × 22 ÷ 3) × (12.5² + 12.5 × 9 + 9²) ≈ 8,058 cm³, which is 8.06 litres of compost to the rim, or roughly 2.13 US gallons.

2. 2Drinking glass

   7.5 cm across the top, 5.5 cm across the base, 13 cm tall gives about 435 cm³, so 0.43 litres to the very brim. Pour to a sensible level and a glass that size is a comfortable 350 ml.

3. 3Lampshade

   40 cm bottom diameter, 30 cm top, 25 cm tall. The slant is √(25² + 5²) ≈ 25.5 cm and the lateral area is π × (20 + 15) × 25.5 ≈ 2,803 cm², so about 0.28 m² of fabric before any seam allowance.

4. 4Hopper outlet

   A silo cone narrowing from 2.4 m to a 0.4 m outlet over 1.8 m holds (π × 1.8 ÷ 3) × (1.2² + 1.2 × 0.2 + 0.2²) ≈ 3.24 m³, which is 3,242 litres of grain sitting below the straight section.

Two checks that prove the formula

A truncated cone sits between two shapes you already trust, and the formula has to land on both of them. Set the top radius equal to the bottom radius and the bracket becomes 3R², the thirds cancel, and the volume is πR²h: a cylinder. Set the top radius to zero instead and the bracket collapses to R², leaving one third of πR²h: a cone. The slant follows the same path, reducing to h in the first case and to √(R² + h²) in the second.

Try both in the calculator above. The diagram redraws into a straight sided cylinder and then into a point, and the note under the answer says which shape you have landed on. If a frustum formula anywhere fails either of those two checks, it is the wrong formula.