Calculator of the truncated square pyramid surface area

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.

Surface Area Formula

To calculate the surface area, first the slant height (apothem) of the lateral face m is determined — the height of the trapezoid forming the lateral surface:

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

Lateral Surface Area:

$$S_{\mathrm{lateral}}=2(a+b)m$$

Total Surface Area:

$$S_{\mathrm{total}}=S_{\mathrm{lateral}}+a^2+b^2$$

where:

• a — larger base side;
• b — smaller base side;
• h — height of the truncated pyramid;
• m — slant height (apothem) of the lateral face.

Please note that these formulas and calculations are intended specifically for a truncated pyramid with a square base.

Where is it applied?

• Construction & Tiling. Cladding monument pedestals, columns, bridge piers, or chimneys requires precise knowledge of the lateral surface area. Calculating the area helps determine the required amount of finishing materials (tiles, stone, plaster) and estimate work costs.
• Hoppers & Industrial Sheathing. To manufacture metal hoppers, tanks, funnel-shaped chutes, smelting furnaces, and industrial containers, it is necessary to calculate the surface area of the parts for cutting sheet metal and determining the weight of the finished product.
• Packaging & Logistics. Many gift boxes and fast-food packaging (such as noodle boxes) have the shape of a truncated pyramid. Calculating the total surface area helps optimize cardboard usage when cutting packaging blanks.
• Thermal & Moisture Insulation. When protecting industrial pyramidal tanks from heat or moisture, you need to know the exact area of the outer walls to purchase waterproofing membranes or insulation panels.
• Landscape Design. Covering artificial mounds with sod, geogrids, or decorative stones requires knowing the exact surface area of the sloped sides of the pyramidal terrace.
• Painting & Protective Coating. For anti-corrosion treatment of supports or painting decorative elements of buildings, the surface area is the key parameter used to calculate the usage of primer and paint.

How to use the calculator?

1. Enter the lower (larger) base side a.
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 decimal rounding precision.
5. Click "Calculate" — the lateral and total surface areas 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.

Do these formulas work for rectangular (non-square) bases?
No, these formulas are intended exclusively for a square base. For a rectangular base, the lateral faces consist of two pairs of different isosceles trapezoids with different slant heights, so the lateral area must be calculated by summing the areas of these four trapezoids individually.

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.