Truncated Pyramid Volume Calculator (Square Frustum)

A truncated pyramid (pyramid frustum) is what you get when you slice off the tip of a pyramid parallel to its base. This calculator handles the square-base case, where the top and bottom are both squares, parallel and centred on the same axis.

V = (h / 3) × (a² + b² + a × b)

Where:

• a = top edge (the smaller square, after truncation)
• b = bottom edge (the larger square, the original base)
• h = vertical height between the two parallel squares

Worked example: the hopper bottom of a square grain bin A grain bin with 4 m × 4 m square cross-section (b = 4 m) has a hopper bottom narrowing to 0.5 m × 0.5 m discharge opening (a = 0.5 m). The hopper is 3 m tall (h = 3 m). V = (3 / 3) × (0.25 + 16 + 2) = 1 × 18.25 = 18.25 m³.

If the bin is filled with wheat (density ~770 kg/m³), the hopper holds about 14 metric tons. That is the quantity still to drain after the straight-sided part above it has emptied.

Worked example: a concrete plug An earth dam’s cross-section is a trapezoid, so its body is a truncated triangular prism rather than this shape. A square frustum does turn up in plug-shaped closures and in bridge-pier footings. Top 6 ft × 6 ft, bottom 18 ft × 18 ft, 30 ft tall. V = (30 / 3) × (36 + 324 + 108) = 10 × 468 = 4,680 ft³.

That is 173 cubic yards, which arrives on about 18 truck loads at the usual 10 yd³ per mixer. Get a delivered price per cubic yard from your own supplier before budgeting, since concrete pricing moves with region and year and the placement cost for a tapered form is usually larger than the concrete itself.

Where truncated pyramids appear in real measurements:

• Hopper bottoms of square or rectangular silos and storage bins.
• Concrete plinths and pedestals for sculpture or column mounting.
• Dam cross-sections (when viewed as a prism with truncated-pyramidal end pieces).
• Pyramidal lampshades that taper to a smaller top opening.
• Architecture. Mesoamerican temples, Mayan and Aztec alike, are built as stacked truncated pyramids.
• Mining and quarry pit estimation. Open-pit mines often approximate as inverted truncated pyramids when calculating excavation volume.
• Filing cabinets and inverted desk lamps with tapered bases.

The Moscow Papyrus, and how old this formula is

Problem 14 of the Moscow Mathematical Papyrus, written around 1850 BCE, works a truncated pyramid with base 4, top 2 and height 6, and gets 56. Run our formula: (6/3) × (4 + 16 + 8) = 2 × 28 = 56. Not an approximation of the modern result, the same result, nearly four thousand years early.

What makes it remarkable is that it is not something you can stumble on by measuring. The scribe had no algebra and left no derivation, and there is still no agreement on how it was found. It is among the oldest non-trivial pieces of mathematics anyone has written down.

Two useful limit cases:

• If a = b there is no taper and the shape is a square prism. The formula gives V = (h/3) × 3b² = b²h, which is the prism formula. ✓
• If a = 0 the top has closed to a point and it is a full pyramid. The formula gives V = (h/3) × b², the square pyramid formula. ✓

Volume vs. average area times height:

A tempting shortcut is V ≈ (area_top + area_bottom) × h / 2. On the hopper above that gives (0.25 + 16) × 1.5 = 24.4 m³ against a true 18.25, so it runs 34% high. The cross term is what the shortcut throws away, and on a steep taper it is most of the answer.

What the ab term actually is

It is the geometric mean of the two end areas, not the middle cross-section. The top area is a², the bottom is b², and √(a² × b²) = ab. Those two are not the same thing: on the hopper, ab = 2 m² while the section halfway up has edge 2.25 m and area 5.06 m².

Both appear in the same identity, which is worth seeing once. Simpson’s rule, or the prismoidal formula, says

V = h × (A_top + 4A_mid + A_bottom) / 6

Put A_mid = ((a+b)/2)² into that and it collapses to h(a² + b² + ab)/3, exactly the frustum formula. So the frustum formula IS the prismoidal formula, and it is exact here because the cross-sectional area of a frustum varies quadratically with height and Simpson’s rule integrates a quadratic exactly.

Sanity check:

• a = b: V = b² × h (square prism). ✓
• a = 0: V = (h/3) × b² (square pyramid). ✓
• h = 0: V = 0. ✓