Truncated Pyramid (Rectangular Frustum) Volume

The volume of a truncated pyramid - a tapered concrete pier or spread-footing pedestal, a rectangular hopper or bin.

Run the calculator

Example

You enter

You get

Checked against published worked examples from: Naval Education and Training Professional Development and Technology Center (U.S. Navy).

Details, formula, and sources

ScopeThe volume of a truncated pyramid - a tapered concrete pier or spread-footing pedestal, a rectangular hopper or bin, or a round-to-rectangular transition. Prismoidal formula V = (h/6)(A1 + A2 + 4 Am), A1 = bottom area, A2 = top area, Am = the section halfway up; exact even when length and width taper at different rates. A 6x6 base, 2x2 top, 4 ft pier takes 69.3 ft^3 = 2.57 yd^3 of concrete; averaging the footprints (a 4x4 mean, 64 ft^3) would short the pour by 5 ft^3. Top 0x0 gives a full pyramid (A1 h/3), equal top/bottom a prism (A1 h). Reported in ft^3, yd^3, and gallons. An offset pyramid, wall thickness, and surface area are separate. A takeoff aid; verify against the drawing.

Formula or tableA1 = Lb x Wb; A2 = Lt x Wt; Am = ((Lb + Lt)/2)((Wb + Wt)/2); V = (h/6)(A1 + A2 + 4 Am). Top 0x0 -> full pyramid (A1 h/3); equal top/bottom -> prism (A1 h).

Source and editionThe truncated-pyramid volume V = (h/6)(A1 + A2 + 4 Am) - the prismoidal formula, standard solid geometry as in Machinery's Handbook (Industrial Press), by name; public domain.

Free accessPure solid geometry, public; the bottom and top dimensions and the height are user-supplied.

What governsEstimate. AHJ and licensed professional govern.

API field namesbottom_length_ft, bottom_width_ft, top_length_ft, top_width_ft, height_ft, volume_ft3, volume_yd3

Related tools