About

A pyramidal frustum is the solid remaining after slicing a pyramid or cone with a plane parallel to its base. Miscalculating its volume leads to material over-ordering in earthwork or concrete pours, directly inflating project costs by 5 - 15%. This calculator applies the exact prismatoid formula V = h3 ⋅ (A₁ + A₂ + √A₁ ⋅ A₂) for polygonal bases and the conical frustum formula for circular cross-sections. It handles square, rectangular, and circular base geometries. Results include lateral surface area, total surface area, slant height, and a real-time 3D wireframe preview.

The tool assumes a right frustum (apex directly above the centroid). Oblique frustums require integral methods not covered here. All computations use IEEE 754 double-precision floating point. Pro tip: for earthwork volume estimates (road embankments, dam cores), always compare the frustum result against the average-end-area method to bound your error. The frustum formula is exact for linear taper; the average-end-area method overestimates by up to 50% on tapered geometries.

Formulas

The general prismatoid formula for the volume of any frustum with parallel polygonal bases:

V = h3 ⋅ (A₁ + A₂ + √A₁ ⋅ A₂)

Where h = perpendicular height between the two parallel bases, A₁ = area of the larger (bottom) base, A₂ = area of the smaller (top) base.

For a square frustum with bottom side a and top side b:

V = h3 ⋅ (a² + b² + ab)

For a circular frustum (truncated cone) with bottom radius R and top radius r:

V = πh3 ⋅ (R² + r² + Rr)

Slant height l for a circular frustum:

l = √h² + (R − r)²

Lateral surface area for a circular frustum:

S_lat = π(R + r) ⋅ l

Total surface area:

S_total = S_lat + A₁ + A₂

For a rectangular frustum with bottom dimensions a × b and top dimensions c × d, the lateral surface is the sum of four trapezoidal faces, each with its own slant height computed from the height and the half-difference of the corresponding parallel edges.

Reference Data

Shape	Volume Formula	Lateral Surface Area	Slant Height	Common Application
Square Frustum	h3(a² + b² + ab)	2(a + b) ⋅ l	√h² + ((a − b) ÷ 2)²	Concrete pedestals, step pyramids
Rectangular Frustum	h3(A₁ + A₂ + √A₁A₂)	Sum of 4 trapezoidal faces	Per-face calculation	Hoppers, bins, chutes
Circular Frustum (Cone)	πh3(R² + r² + Rr)	π(R + r) ⋅ l	√h² + (R − r)²	Buckets, lampshades, cooling towers
Hexagonal Frustum	h3(A₁ + A₂ + √A₁A₂)	Sum of 6 trapezoidal faces	Per-face	Architectural columns
Triangular Frustum	h3(A₁ + A₂ + √A₁A₂)	Sum of 3 trapezoidal faces	Per-face	Structural supports
Earthwork Prismoid	Simpson: h6(A₁ + 4A_m + A₂)	N/A (irregular)	N/A	Road embankments
Avg-End-Area	h2(A₁ + A₂)	N/A	N/A	Quick field estimate
Pyramid (full)	13A_base ⋅ h	Depends on base shape	Apex to base-edge midpoint	Limit case (A₂ = 0)
Prism (cylinder)	A_base ⋅ h	Perimeter ⋅ h	h	Limit case (A₁ = A₂)
Great Pyramid of Giza	≈ 2,583,283 m³	≈ 85,000 m²	186.4 m	Base 230.4 m, height 146.5 m
Concrete Density	2,400 kg/m³ (normal weight). Multiply volume by density for mass estimation.
Steel Density	7,850 kg/m³. Used for frustum-shaped castings and molds.
Water Density	1,000 kg/m³ at 4°C. Useful for tank capacity in frustum-shaped vessels.

Frequently Asked Questions

The prismatoid formula V = (h/3)(A₁ + A₂ + √(A₁·A₂)) is mathematically exact for any frustum with linearly tapering sides. The average-end-area method V = (h/2)(A₁ + A₂) always overestimates volume because it treats the solid as a prism. The overestimation error can reach 50% when A₁ and A₂ differ greatly. For earthwork where A₁ ≈ A₂, both methods converge, but for sharply tapered frustums the prismatoid formula is strictly required.

When A₂ = 0, the frustum formula reduces to V = (h/3)·A₁, which is the standard pyramid (or cone) volume formula. This calculator handles that degenerate case correctly. The slant height becomes the true slant height from base to apex.

No. This calculator assumes a right frustum where the axis connecting the centroids of both bases is perpendicular to both base planes. For oblique frustums, the volume formula remains valid (Cavalieri's principle), but the lateral surface area calculation requires integration over the actual surface geometry. The slant height reported here would not be physically meaningful for an oblique frustum.

Multiply the computed volume by the material density. Concrete: approximately 2,400 kg/m³. Steel: 7,850 kg/m³. Water: 1,000 kg/m³ at 4°C. Soil (compacted): 1,600-2,100 kg/m³ depending on moisture content. Always use the density of the material in its placed condition, not its dry laboratory density.

A rectangular frustum has two pairs of trapezoidal lateral faces. Each pair has a different slant height because the taper along the length dimension differs from the taper along the width dimension. The slant height along the length faces is √(h² + ((a−c)/2)²) and along the width faces is √(h² + ((b−d)/2)²). This calculator reports both values separately.

The prismatoid formula used here is consistent with ASTM C29 (bulk density calculations) and ISO 12491 geometric measurement principles. For regulatory compliance in earthwork, refer to AASHTO guidelines which accept both the prismatoid formula and the average-end-area method with correction factors.