Calculator of the truncated square pyramid volume

What is a Truncated Pyramid?

A truncated pyramid (frustum of a pyramid) is a geometric solid formed by cutting off the top part of a pyramid with a plane parallel to the base. It has two square bases of different sizes and four trapezoidal lateral faces.

Volume Formula

The volume of a truncated square pyramid is calculated using the following formula:

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

where:

• a — larger base side;
• b — smaller base side;
• h — height of the truncated pyramid.

It is worth noting that the formulas and this online calculator are designed specifically for a truncated pyramid with a square base.

Apothem of Lateral Face

Additionally, the calculator computes the slant height (apothem) of the lateral face m — the height of the trapezoid forming the lateral surface:

$$m=\sqrt{h^2+{\left(\frac{a-b}{2}\right)}^2}$$

Where is it applied?

• Construction & Architecture. Columns, monument pedestals, bridge piers, and towers often have the shape of a truncated pyramid. Calculating the volume helps determine the exact amount of concrete, brick, or other building materials needed during the design stage.
• Manufacturing & Industrial Hoppers. Many storage bins for bulk materials, grain silos, funnels, smelting furnaces, and industrial containers are designed as inverted truncated pyramids. Knowing the exact volume is essential for estimating their capacity and preventing overflow.
• Confectionery & Cooking. Baking molds, chocolate candies, and traditional festive desserts often take the shape of a truncated pyramid. Volume calculation allows chefs to portion ingredients precisely.
• 3D Modeling & Computer Graphics. In computer vision and graphics, the volume of the camera's visible area (known as the view frustum) is used to determine which objects fall within the frame and need rendering.
• Landscape Design. When creating terraces, artificial mounds, raised beds, or decorative garden elements in the shape of pyramids, you need to know the volume of soil or sand required to build them.
• Geology & Excavation. During excavations, trenching, or pit digging, soil piles or holes of a truncated pyramid shape are frequently formed. Calculating their volume is important for estimating transportation costs for soil removal.

How to use the calculator?

1. Measure or take the lower (larger) base side a from the drawing.
2. Enter the upper (smaller) base side b (b < a).
3. Enter the height h (or the apothem m, depending on the selected calculation option).
4. Select the desired decimal precision.
5. Click "Calculate" — the volume and slant height will be computed instantly.

Frequently Asked Questions (FAQ)

What is the difference between a regular pyramid and a truncated pyramid?
A regular pyramid has a single base and tapers to a single vertex at the top. A truncated pyramid (frustum) is formed when the top portion of a regular pyramid is cut off by a plane parallel to the base. As a result, it features two parallel bases (lower and upper) instead of one base and a vertex.

Does this formula work for rectangular (non-square) bases?
No, the formula and the online calculator on this page are specifically for a truncated pyramid with a square base (where all four sides of each base are equal). For a rectangular base, the volume formula is different because it must account for both the length and width of the two bases: V = h/3 * (S1 + √(S1 * S2) + S2), where the areas of the bases are S1 = a1 * b1 and S2 = a2 * b2.

What is a frustum in geometry?
In geometry, a frustum is the portion of a solid (usually a cone or a pyramid) that lies between one or two parallel planes cutting through it. Most commonly, it refers to the shape obtained by cutting off the apex of a solid parallel to its base.