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Prismatoid Volume Calculator

Enter the bottom, mid-section and top areas plus the height to get the prismatoid volume.

How it is calculated

The tool applies the prismoidal formula V = h/6 × (A₁ + 4·Aₘ + A₂). Here A₁ is the area of the bottom face, A₂ the area of the top face, Aₘ the area of the cross-section cut exactly halfway between them, and h the perpendicular distance between the two end faces. Areas must be zero or more and the height must be greater than zero; otherwise nothing is returned. A zero end area is accepted, which is how a pyramid or a cone (pointed top) is entered. The tool does not convert units: if the areas are in m² and the height in m, the volume is in m³, rounded to two decimals. The mid-section is the one at half height, not the average of the two end areas, and that is what makes the formula exact for many tapering shapes.

Worked example

An excavation is 10 m deep. The area at the bottom is 120 m², at mid-depth 90 m², and at the top 64 m². Step 1: weight the middle area four times, 120 + 4 × 90 + 64 = 544. Step 2: multiply by h/6 = 10/6. Step 3: 544 × 10 / 6, which the tool shows as 906.67. In these units that is 906.67 m³ of material.

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Common mistakes

  • Entering the average of the two end areas as the mid-section. Aₘ must be the area measured at half height; for a tapering shape it is not (A₁ + A₂)/2, and the answer will be wrong.
  • Using the slanted side length as the height. The height h is the perpendicular distance between the two parallel end faces.
  • Mixing units, such as areas in m² and the height in cm. Convert everything to the same length unit first, because the result is simply that unit cubed.
  • Entering a dimension (a side or a radius) instead of an area. All three area fields expect areas, so for a circular section enter π × r², not r.
  • Entering 0 for a pointed end and then thinking the tool failed: 0 is accepted for A₁, Aₘ or A₂, and a pyramid with base 100, mid-area 25, top 0 and height 12 gives 400, the same as one third of base times height.

About this calculator

The prismoidal formula, V = h⁄6 × (A₁ + 4Aₘ + A₂), computes the volume of a prismatoid — any solid whose cross-sectional area varies smoothly between two parallel end faces — from the area of the bottom face A₁, the area of the cross-section exactly at mid-height Aₘ, the area of the top face A₂, and the perpendicular height h between the two ends.

Because it's essentially Simpson's Rule applied to volume, this single formula gives the exact volume for a remarkable range of shapes — prisms, pyramids, cones, frustums, wedges, spheres and even cylinders — as long as the cross-sectional area varies as a polynomial of degree three or less along the height. Civil engineers use it constantly for earthwork and excavation volumes, and it appears in surveying, mining and construction estimating wherever an irregular but smoothly-tapering solid needs a precise volume.

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Comments (1)

  • Editorial team

    The middle area must be measured at exactly half the height, not averaged from the two ends. A quick test: a pyramid with a 100 base area, 0 top area and a height of 6 has a mid-height section of 25 (a quarter of the base), and the tool returns 200, which equals base times height divided by 3. If your mid-section comes out very different from that kind of pattern, recheck where you cut. Which solids are you using this for?

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