HesapHanem

Pyramid Frustum Volume Calculator

Volume of a truncated pyramid.

Result

224.0000

How it is calculated

This tool treats both faces of the frustum as squares. Enter the bottom edge a1, the top edge a2 and the perpendicular height h between the two faces, all in one unit. First it squares the edges to get the face areas A₁ = a1² and A₂ = a2². Then V = h / 3 × (A₁ + A₂ + √(A₁ × A₂)), rounded to 4 decimals. The square root term equals a1 × a2. If a1 = a2 the shape is a prism and the formula returns a² × h. Zero, negative or missing values return no result, and the tool does not require a1 to be the larger edge.

Worked example

A square frustum has bottom edge 8, top edge 4 and height 6. Face areas: A₁ = 64 and A₂ = 16. The square-root term is √(64 × 16) = √1024 = 32. Add the three parts: 64 + 16 + 32 = 112. Then multiply by h / 3 = 6 / 3 = 2: 2 × 112 = 224.0000 cubic units. Check: if the missing pyramid tip were included, the full pyramid would be 12 tall, and the 224 equals the big pyramid minus the small one.

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Common mistakes

  • Entering the face areas in a1 and a2 instead of the edge lengths. The tool squares the inputs itself, so areas would be squared twice.
  • Using the slanted side of the trapezoidal face as the height. The formula needs the perpendicular distance between the two parallel faces.
  • Using the tool for a round frustum (truncated cone) or a rectangular one with two different edge lengths per face. Both faces here are squares.
  • Averaging the two face areas and multiplying by height. That gives 40 × 6 = 240, which is too big; the pyramid shape needs the extra term and the division by 3.

About this calculator

A frustum is what remains of a pyramid after its top is sliced off by a plane parallel to the base — a shape with two similar parallel faces (typically squares) of different sizes joined by trapezoidal sides. Its volume is given by V = (h/3)(A₁ + A₂ + √(A₁·A₂)), where A₁ and A₂ are the areas of the top and bottom faces and h is the perpendicular height between them; for a square frustum this reduces to a formula using just the two edge lengths and the height.

This is actually one of the oldest surviving pieces of applied mathematics: the Moscow Mathematical Papyrus (c. 1850 BCE) contains the earliest known correct calculation of a frustum’s volume, likely developed for estimating grain storage or building materials in ancient Egypt. Today the same formula shows up wherever a truncated-pyramid shape is built for real: architects use it for hip-roof and tapered-tower volumes, engineers use it to size hoppers and grain silos, and manufacturers use it to compute material volume for truncated-pyramid molds and packaging.

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