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 — both top and bottom are squares, parallel and centered.

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 — 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 of wheat — which is the amount left to drain after the cylindrical part above is empty.

Worked example — concrete dam cross-section: A small earth dam with trapezoidal cross-section is a truncated triangular prism — but a SQUARE-BASE truncated pyramid appears in plug-shaped concrete dam closures. 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’s 173 cubic yards of concrete. At ~$200 per cubic yard delivered: $34,600 in concrete alone.

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.
• Truncated pyramid frustums in architecture — Mesoamerican temples (Mayan, Aztec) 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 Egyptian “two-third” estimate:

An old Egyptian rule for truncated pyramid volume (preserved in the Moscow Papyrus, c. 1850 BCE) was V ≈ (h / 3) × (a² + b² + ab), which is EXACTLY the modern formula. Egyptian surveyors knew this 4,000 years ago for taxing land and counting grain.

This is one of the oldest non-trivial mathematical results in human history.

Two useful limit cases:

• If a = b (no truncation): V = (h / 3) × 3b² = b² × h. This is wrong for a true pyramid AND wrong for a true prism. Wait — let me reconsider.

  When a = b, the shape is actually a square prism (rectangular box). The truncated-pyramid formula gives V = (h/3) × 3b² = b²h. That matches the prism formula. ✓

• If a = 0 (full pyramid, no truncation): V = (h / 3) × b² = b²h / 3. Matches the square pyramid formula. ✓

Volume vs. average area times height:

A common rule-of-thumb approximation is V ≈ (area_top + area_bottom) × h / 2. For our hopper example: (0.25 + 16) × 1.5 = 24.4 m³ — too high by 34%. The actual formula’s “ab cross term” matters a lot for tapered shapes.

The cross-term ab is geometrically the area of an intermediate cross-section halfway up the frustum.

Sanity check:

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