Pyramid Frustum Volume Calculator
Find the volume of a truncated square pyramid (frustum) from its two square bases and height, with slant height and surface area.
How it works
Volume = (h ÷ 3) · (A₁ + A₂ + √(A₁·A₂)) · A = side² · Slant ℓ = √(h² + ((a − b) ÷ 2)²) · Lateral = 4 · ((a + b) ÷ 2) · ℓ
A pyramid frustum is a square pyramid with its top cut off parallel to the base — the shape of a monitor stand, a hopper or the Louvre pyramid’s truncated cousins. The prismatoid formula averages the two base areas together with the geometric-mean term √(A₁·A₂), a blend that captures how the cross-section shrinks smoothly from bottom to top. Each of the four trapezoidal faces slopes along the slant height √(h² + ((a − b) ÷ 2)²).
Worked example
For a bottom side of 6, top side of 3 and height 8: A₁ = 36, A₂ = 9, and √(36 × 9) = 18, so volume = (8 ÷ 3) × (36 + 9 + 18) = (8 ÷ 3) × 63 = 168 cubic units. The slant height is √(8² + 1.5²) = √66.25 ≈ 8.1394, the four faces add 4 × 4.5 × 8.1394 ≈ 146.509, and the total surface area is 36 + 9 + 146.509 ≈ 191.509.
Sources: Frustum — Wikipedia
Frequently asked questions
Why include the √(A₁·A₂) term?
It is the geometric mean of the two end areas, and it is exactly what the prismatoid (Simpson) rule requires to integrate the shrinking cross-sections correctly. Averaging only A₁ and A₂ would overstate the volume; the middle term corrects it.
Does this work for a full pyramid?
Yes — set the top side b to zero and the formula reduces to (h ÷ 3) × A₁, the ordinary square-pyramid volume. The frustum result is the general case.
Is the surface area only valid for square bases?
The volume formula works for any similar top and bottom, but the slant-height and surface-area figures here assume both ends are squares centred over each other. Rectangular or off-centre tops need face-by-face treatment.
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This calculator is for educational purposes. Double-check important results before acting on them.