Truncated Square Pyramid — Definition, Formula & Examples

A truncated square pyramid is the solid that remains when the top of a square pyramid is sliced off by a plane parallel to the base. It has two parallel square faces (a larger base and a smaller top) connected by four trapezoidal lateral faces.

A truncated square pyramid (or square frustum) is a frustum formed by intersecting a right square pyramid with a plane parallel to its base, producing a solid bounded by two square faces of different side lengths and four congruent isosceles trapezoids.

Key Formula

V = \frac{h}{3}\left(a^2 + ab + b^2\right)

Where:

$V$ = Volume of the truncated square pyramid
$h$ = Perpendicular height (distance between the two square faces)
$a$ = Side length of the larger base square
$b$ = Side length of the smaller top square

Worked Example

Problem: Find the volume of a truncated square pyramid with a base side length of 6 cm, a top side length of 2 cm, and a height of 9 cm.

Substitute into the formula: Use the volume formula with a = 6, b = 2, and h = 9.

V = \frac{9}{3}\left(6^2 + 6 \cdot 2 + 2^2\right)

Simplify inside the parentheses: Compute each term: 36 + 12 + 4 = 52.

V = 3 \times 52

Calculate the volume: Multiply to get the final result.

V = 156 \text{ cm}^3

Answer: The volume is 156 cm³.

Why It Matters

Truncated pyramids appear in architecture, packaging, and engineering — any context where a tapered shape has its tip removed. Mastering this formula prepares you for composite-solid volume problems on geometry exams and standardized tests like the SAT and ACT.

Common Mistakes

Mistake: Using the average of the two base areas instead of the prismoidal formula.

Correction: The correct formula includes three terms — a², ab, and b² — divided by 3, not just the average of the two areas. Simply averaging the bases underestimates the volume.