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Prismoidal Volume Calculator

Enter the two end areas, the middle area, and the length to find the prismoidal volume.

How it is calculated

The prismoidal formula needs three cross-sectional areas and the distance between the end sections: the area at the start A1, the area measured exactly halfway along Am, the area at the end A2, and the length L. The volume is V = (L / 6) × (A1 + 4 × Am + A2). The tool treats the inputs as square metres and metres, so the volume is in m³. It also converts that to cubic yards by dividing by 0.764555, and reports the mean section area V / L. If you work in feet, enter feet and read the first result as ft³; the yd³ line then no longer applies. All four inputs must be above zero. A section that tapers to a point cannot be entered as 0, so use a very small area instead.

Worked example

A 30 m long cutting has end areas of 42.5 m² and 55.5 m², and a measured mid-section of 48 m². Weighted sum: 42.5 + 4 × 48 + 55.5 = 42.5 + 192 + 55.5 = 290. Volume: (30 / 6) × 290 = 5 × 290 = 1,450 m³. The tool shows 1,450.000 m³, which is 1,896.53 yd³, with an average section of 48.33 m². The simple average of the two ends would give 30 × 49 = 1,470 m³, about 20 m³ more, because it ignores the mid-section.

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Common mistakes

  • Using the average of the two end areas as the middle area. Am must be the area measured at the halfway point, otherwise the formula collapses to a cruder average-end-area estimate.
  • Measuring the middle section at the wrong place. It has to sit exactly halfway between the ends, not at a convenient station.
  • Mixing units, such as areas in square feet and length in metres. The labels assume metres, so keep every input in the same metric or consistent unit.
  • Entering 0 for an end that comes to a point, such as the apex of a mound. The tool rejects zero areas, so give a tiny positive value for that end.
  • Using it across a long stretch where the shape changes abruptly. One formula run over a very long length can miss that change, so split the job into shorter sections and add the volumes.

About this calculator

Civil engineers computing earthwork — cut-and-fill volumes for roads, dams and excavations — often need the volume of a solid between two parallel cross-sections that aren't uniform in shape. The prismoidal formula, V = (L/6)(A1 + 4·Am + A2), estimates that volume from the two end areas (A1, A2), the area at the exact midpoint (Am), and the distance L between the ends.

The formula is a direct application of Simpson's rule for numerical integration, and it is exact for any solid whose cross-sectional area varies as a quadratic (or lower-order) function of position — which covers prisms, pyramids, cones, wedges and most graded roadway sections. Because it uses a middle sample instead of just averaging the two ends, it is noticeably more accurate than the simpler average-end-area method commonly used for quick grading estimates.

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Comments (1)

  • Editorial team

    The mid-section area has to be measured at the halfway point, not averaged from the ends. If you type the mean of A1 and A2 as Am, the formula collapses to the plain average-end-area answer and you lose the benefit. The prismoidal form is worth using when the section changes unevenly along the length. Do you use it for cut and fill, stockpiles or something else?

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