Square Pyramid Frustum Calculator

What is a Square Pyramid Frustum?

A square pyramid frustum (also called a truncated square pyramid) is the three-dimensional shape you get when you slice off the top of a regular square pyramid with a cut parallel to the base. The resulting solid has two parallel square faces -- a larger bottom square (side $a$) and a smaller top square (side $b$) -- connected by four trapezoidal lateral faces. This shape appears in architecture (Mayan and Aztec stepped pyramids, ziggurats, plinths), everyday objects (tapered planters, lampshades, buckets), and engineering (hoppers, transition sections). For related shapes, try our Pyramid Calculator and Conical Frustum Calculator.

This calculator supports four solve modes: Volume, Total Surface Area, Lateral Surface Area, and Height. You can input dimensions in meters, centimeters, inches, feet, or yards, and volume results display in cubic meters, liters, gallons, or cubic feet.

Square Pyramid Frustum Formulas

For a square pyramid frustum with bottom edge $a$, top edge $b$, and perpendicular height $h$:

Face Slant Height: $\ell = \sqrt{h^2 + \left(\frac{a - b}{2}\right)^2}$

The slant height is the perpendicular distance from the midpoint of a bottom edge up the trapezoidal face to the midpoint of the corresponding top edge. It is used in the lateral surface area formula.

Corner Slant Edge: $e = \sqrt{h^2 + \frac{(a - b)^2}{2}}$

The slant edge is the actual length of a corner-to-corner edge connecting a bottom vertex to the corresponding top vertex.

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

The volume formula is a specialization of the general frustum formula $V = \frac{h}{3}\left(A_{\text{bottom}} + \sqrt{A_{\text{bottom}} \cdot A_{\text{top}}} + A_{\text{top}}\right)$. For two squares with sides $a$ and $b$, the geometric mean $\sqrt{a^2 \cdot b^2} = ab$, giving the simplified form. When $b \to 0$ the formula collapses to $\frac{1}{3}a^2h$ (a full square pyramid); when $b \to a$ it becomes $a^2h$ (a square prism).

Lateral Surface Area: $S_{\text{lat}} = 2(a + b)\ell$

The lateral surface area covers only the four trapezoidal faces.

Total Surface Area: $S = 2(a + b)\ell + a^2 + b^2$

The total surface area includes both square ends plus the four trapezoidal faces.

Height from Volume: $h = \frac{3V}{a^2 + ab + b^2}$

When you know the volume and both edge lengths, this inverse formula gives the required height.

How to Use the Calculator

Select Solve Mode: Choose Volume, Total Surface Area, Lateral Surface Area, or Height.
Enter Bottom Edge (a): The side length of the larger (bottom) square face.
Enter Top Edge (b): The side length of the smaller (top) square face. The two edges can be in either order -- the calculator works for both $a > b$ and $b > a$.
Enter Height (h) or Volume (V): Depending on the solve mode, enter the perpendicular height or the target volume.
Select Units: Choose length units (m, cm, in, ft, yd) and volume units (m³, L, gal, ft³) as needed.

Example Calculation

Example: Find the volume, slant height, lateral surface area, and total surface area of a square pyramid frustum with bottom edge $a = 6$ m, top edge $b = 2$ m, and height $h = 4$ m.

Slant height: $\ell = \sqrt{4^2 + ((6 - 2)/2)^2} = \sqrt{16 + 4} = \sqrt{20} \approx 4.4721$ m
Volume: $V = (4/3)(36 + 12 + 4) = (4/3)(52) = 208/3 \approx 69.33$ m³
Lateral surface area: $S_{\text{lat}} = 2(6 + 2)(4.4721) = 16 \times 4.4721 \approx 71.55$ m²
Total surface area: $S = 71.55 + 36 + 4 \approx 111.55$ m²

Applications

Stepped Pyramids: Mayan, Aztec, and Mesopotamian ziggurats are built from stacked frustum courses. Total volume is the sum of each course's frustum volume.
Architectural Plinths: Tapered square-section base blocks and finials are frustum-shaped.
Planters: Tapered square flowerpots and planter boxes have frustum geometry for capacity calculations.
Lampshades: Square-section lampshade frames need lateral surface area for fabric estimation.
Hoppers: Transition sections with square inlet and outlet use frustum volume and slant for sheet-metal layout.
Packaging: Tapered crates and containers use frustum formulas for material takeoff and cube efficiency.

Frequently Asked Questions

How do you calculate the volume of a truncated square pyramid?

Use $V = (h/3)(a^2 + ab + b^2)$, where $a$ is the bottom edge, $b$ is the top edge, and $h$ is the perpendicular height. For example, with $a = 6$ m, $b = 2$ m, $h = 4$ m: $V = (4/3)(36 + 12 + 4) = 208/3 \approx 69.33$ m³.

What is the difference between slant height and slant edge?

Slant height $\ell = \sqrt{h^2 + ((a - b)/2)^2}$ is the perpendicular distance from the midpoint of a bottom edge up the trapezoidal face to the midpoint of the top edge. Slant edge $e = \sqrt{h^2 + (a - b)^2/2}$ is the actual corner-to-corner edge length. Lateral surface area uses $\ell$, not $e$.

What happens when the top and bottom edges are equal?

When $a = b$, the frustum becomes a square prism. The slant height equals the height ($\ell = h$), the lateral surface is four rectangles, and the volume simplifies to $V = a^2 h$. The formulas handle this case naturally.

What happens when the top edge is zero?

When $b = 0$, the frustum collapses into a full square pyramid. The formulas become $V = (1/3)a^2 h$, $\ell = \sqrt{h^2 + (a/2)^2}$, $S_{\text{lat}} = 2a\ell$, and $S = S_{\text{lat}} + a^2$. The results match the regular square pyramid formulas exactly.

Where are pyramid frustums used in real life?

Square pyramid frustums appear in stepped pyramids (Mayan, Aztec, ziggurats), architectural plinths and finials, tapered planter boxes, square lampshades, hoppers and transition sections in HVAC and material handling, and tapered packaging crates. Any time a shape narrows from a larger square base to a smaller square top, the result is a frustum.

How does this relate to the cone frustum formula?

Both are specializations of the same general formula $V = (h/3)(A_b + \sqrt{A_b \cdot A_t} + A_t)$. For cones (circular faces), $A = \pi r^2$ giving $V = (\pi h/3)(R^2 + Rr + r^2)$. For square frustums (square faces), $A = a^2$ giving $V = (h/3)(a^2 + ab + b^2)$. The structural pattern is identical.

Report