Pyramid Frustum Calculator

Pyramid Frustum (Truncated Square Pyramid)

A pyramid frustum is what is left when you slice the top off a pyramid with a cut parallel to the base. This calculator handles the square version, where the top and bottom faces are both squares.

Formulas:

The volume formula is the one worth pausing on. It is not the average of the two end areas times the height, which is the guess almost everyone makes first. That would give (h/2)(a² + b²), and on the worked example below it lands at 43.5 instead of 39, about 12% too high. The extra ab term, and the division by 3 rather than 2, are what account for the sloping sides.

Variables:

• a = top side length (smaller square)
• b = bottom side length (larger square)
• h = perpendicular height between the two bases
• l = slant height of each trapezoidal face

Worked example, a stepped pedestal (a = 2 m, b = 5 m, h = 3 m):

• Slant height: l = √(9 + 2.25) = √11.25 ≈ 3.354 m
• Volume: (3/3)(4 + 10 + 25) = 39 m³
• Lateral SA: 2(2+5)(3.354) = 14 × 3.354 ≈ 46.96 m²
• Total SA: 4 + 25 + 46.96 ≈ 75.96 m²

Type those three numbers into the boxes above and that is what comes back.

Special cases:

• If a = 0 the frustum is a complete pyramid, and the formula collapses to V = hb²/3
• If a = b it is a square prism, and the formula collapses to V = ha²
• Both fall straight out of the prismatoid formula, V = (h/6)(A_top + 4A_mid + A_bot) with A_mid = ((a+b)/2)²

Those two cases are worth testing yourself. Set a = 0 with b = 5 and h = 3 and you get 25 m³, exactly a third of the 75 m³ box that would enclose it. Set a = 5 as well and you get the full 75.

Real-world pyramid frustums: The base of the Washington Monument is a frustum shape. Ancient Mesoamerican step pyramids, the bases of obelisks, tapered building columns, and some crystal structures all follow this geometry. The frustum also appears in architecture when a tower narrows as it rises.