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Pyramid Frustum Calculator

Calculate the volume of a square or triangular pyramid frustum (truncated pyramid) using top side, bottom side, and height.

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What is a Pyramid Frustum?

A pyramid frustum is the portion of a pyramid that remains after the top part has been cut off by a plane parallel to the base. The resulting shape has two parallel polygonal faces: a smaller top face and a larger bottom face, connected by trapezoidal lateral faces. A frustum is also known as a truncated pyramid.

Pyramid frustums appear in many real-world objects including buckets, flower pots, lampshades, the Great Pyramid of Giza (which is technically a frustum now as it has lost its apex), and various architectural and engineering structures. Understanding frustum geometry is essential for construction, manufacturing, and design.

This calculator helps you compute the volume, surface areas, and slant height of a pyramid frustum. You can choose between square (4 sides) and triangular (3 sides) frustums.

Pyramid Frustum Formulas

For a pyramid frustum with top side length a, bottom side length b, and height h:

Top Area (Square): A_top = a²

Top Area (Triangular): A_top = (√3/4)a²

Bottom Area (Square): A_bot = b²

Bottom Area (Triangular): A_bot = (√3/4)b²

Volume: V = (1/3)h(A_top + A_bot + √(A_top × A_bot))

This is the universal frustum volume formula, where A_top and A_bot are the areas of the top and bottom faces respectively.

Slant Height: s = √(h² + ((b - a)/2)²)

The slant height is the distance along the face from the top edge to the bottom edge.

Lateral Surface Area: L = n × ((a + b)/2) × s

Where n is the number of sides (3 for triangular, 4 for square). Each lateral face is a trapezoid.

Total Surface Area: A = A_top + A_bot + L

The total surface area includes all exposed faces: top, bottom, and lateral.

How to Use the Calculator

  1. Select Shape: Choose between a square or triangular pyramid frustum.
  2. Enter Top Side Length (a): The side length of the top (smaller) face.
  3. Enter Bottom Side Length (b): The side length of the bottom (larger) face.
  4. Enter Height (h): The perpendicular distance between the top and bottom faces.

Example Calculation

Example: Find the volume of a square pyramid frustum with top side a = 4 cm, bottom side b = 8 cm, and height h = 10 cm.

  • Top Area: A_top = 4² = 16 cm²
  • Bottom Area: A_bot = 8² = 64 cm²
  • Volume: V = (1/3) × 10 × (16 + 64 + √(16 × 64)) = (1/3) × 10 × (16 + 64 + 32) = (1/3) × 10 × 112 = 373.33 cm³

Also check: Pyramid Calculator, Cone Calculator, Cylinder Calculator, Volume Calculator, Surface Area Calculator, and Tank Volume Calculator.

Frequently Asked Questions

What is the difference between a frustum and a truncated pyramid?

A frustum and a truncated pyramid are the same thing. The term "frustum" comes from Latin meaning "a piece broken off," and it refers to the portion of a pyramid (or cone) that remains after removing the top portion with a cut parallel to the base. The terms are used interchangeably in geometry.

Can the top side be larger than the bottom side?

Yes, this is called an inverted frustum where the top is wider than the bottom. The calculator handles this case correctly since the formulas use the square of the side difference for slant height calculations, which works regardless of which side is larger.

What happens when the top and bottom sides are equal?

When the top side (a) equals the bottom side (b), the frustum becomes a standard prism. The lateral faces become rectangles instead of trapezoids, the slant height equals the height, and volume becomes V = A_bot × h (area of base times height).

Where are pyramid frustums used in real life?

Pyramid frustums appear in many everyday objects: buckets, flower pots, lampshades, planting containers, and various architectural elements. In engineering, they are used in hoppers, funnels, and structural transitions. The Great Pyramid of Giza is a famous example of a square pyramid frustum.

How is a pyramid frustum different from a conical frustum?

A pyramid frustum has polygonal top and bottom faces (triangular or square) with flat trapezoidal lateral faces, while a conical frustum has circular top and bottom faces with a curved lateral surface. A conical frustum is essentially the limit of a pyramid frustum as the number of sides approaches infinity.

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