Square Pyramid Frustum Calculator (Truncated Square Pyramid)

When to Use Each Variable

• Solve for Volume — when a, b, and h are known and you need the capacity of a planter box, stepped-pyramid course, hopper section, or tapered packaging block.
• Solve for Total Surface Area — when you need the entire exterior including both square ends — common for finished plinths and closed architectural blocks.
• Solve for Lateral Surface Area — when only the four slanted trapezoidal sides matter — lampshade frames, planter side wraps, sheet metal for an open-top hopper.
• Solve for Height — when both square edges are fixed by available stock and you need the height that gives a target volume.

Key Concepts

A square pyramid frustum is what you get when you slice a regular square pyramid with a plane parallel to its base and remove the small pyramid on top — the same silhouette you see on Mayan and Aztec stepped pyramids, ziggurats, and architectural plinths. Three numbers determine the geometry: the bottom edge a, the top edge b, and the perpendicular height h. There are two distinct slant measurements: the face slant height ℓ = √(h² + ((a − b) / 2)²) — the perpendicular distance from a face midpoint up to the top edge midpoint, used in the lateral-area formula — and the slant edge e = √(h² + (a − b)² / 2) — the actual length of the corner-to-corner edge. The volume V = (h / 3)(a² + a b + b²) is a specialization of the general frustum formula V = (h / 3)(A_bottom + √(A_bottom · A_top) + A_top): for two squares with sides a and b the geometric mean √(A_bottom · A_top) = √(a² · b²) = a b, so the bracketed sum collapses to a² + a b + b². The volume interpolates smoothly between two familiar cases: when b → 0 it becomes (1/3) a² h (a full square pyramid); when b → a it becomes a² h (a square prism).

Applications

• Mayan, Aztec, and ziggurat stepped pyramids — each course of a stepped pyramid is a frustum, and total volume is the sum of frustum volumes per course
• Architectural plinths, finials, and tapered base blocks — square-section transitions that are calculated as frusta during stonework or metal layout
• Planter boxes and tapered flowerpots with square cross-section — capacity from rim and base edges and depth
• Lampshade frames with square cross-section — fabric needed is the lateral surface area
• Hoppers and transition sections with square inlet and outlet — sized using the frustum volume and slant for sheet-metal layout
• Packaging and shipping crates with tapered profiles — material takeoff and cube efficiency calculations
• Geometry instruction — the textbook shape that connects square-pyramid and square-prism formulas

Common Mistakes

• Using the slant edge √(h² + (a − b)² / 2) in the lateral-area formula instead of the face slant height √(h² + ((a − b) / 2)²) — the two are different by a factor involving the corner geometry, and only the face slant ℓ multiplies the (a + b) sum in S_lat
• Forgetting the a b mixed term in the volume formula — V = (h / 3)(a² + b²) is wrong; the correct sum is a² + a b + b² (the geometric mean of the two square areas)
• Computing slant as √(a² + h²) (the square-pyramid formula with apex over center) instead of √(h² + ((a − b) / 2)²) — the offset (a − b) / 2 is what matters once the top is cut off, not the half-base a / 2
• Including only one square end disk in the total surface area — both the bottom a² and the top b² count when the frustum is closed at both ends
• Applying the cone-frustum formula V = (π h / 3)(R² + R r + r²) directly with a and b in place of R and r — there is no π for a square cross-section; the correct form is V = (h / 3)(a² + a b + b²)