Pyramid Frustum Volume Calculator
A pyramid frustum is what remains when the top of a pyramid is sliced off by a plane parallel to its base, leaving two parallel faces of different sizes joined by sloping sides. Its volume is found from the areas of the two parallel faces and the perpendicular height between them. This calculator takes those three numbers and returns the volume using the prismatoid formula for a frustum: the height divided by three, multiplied by the sum of the top area, the bottom area and the square root of their product. The square-root term is the geometric mean of the two areas, and it captures how the cross-section tapers smoothly between the faces. The formula applies to any frustum whose faces are similar shapes, whether square, rectangular, triangular or otherwise, as long as you supply the actual face areas. Builders, civil engineers, students and anyone estimating the volume of a tapered hopper, a planter, an embankment or a lampshade use it. Enter the two face areas and the height to get the volume immediately, with all inputs editable. Every figure here is computed deterministically from the formula shown below, with a worked example that reconciles exactly to the calculator so you can follow each step yourself.
A frustum's volume is h/3 (A1 + A2 + sqrt(A1 A2)). For areas 100 and 36 with height 9, the volume is 588.00.
Pyramid Frustum Volume formula
V = (h/3) (A1 + A2 + sqrt(A1 x A2))
V = volume
A1 = area of one parallel face
A2 = area of the other parallel face
h = perpendicular height between faces
The square-root term is the geometric mean of the two face areas. Adding it to the sum of the two areas and multiplying by one third the height accounts for the smooth taper of the solid.
Worked example
Find the volume of a frustum with a bottom face area of 100, a top face area of 36 and a height of 9.
- Geometric mean = sqrt(100 x 36) = sqrt(3600) = 60
- Sum of terms = 100 + 36 + 60 = 196
- V = (9 / 3) x 196 = 3 x 196 = 588.00
The frustum volume is 588.00. These are the calculator's default inputs, so the result above matches the widget exactly.
Volume for height 9 at various face areas
V = (9/3)(A1 + A2 + sqrt(A1 A2)).
| A1 | A2 | Volume |
|---|---|---|
| 100 | 36 | 588.00 |
| 100 | 25 | 465.00 |
| 64 | 16 | 336.00 |
| 50 | 50 | 450.00 |
| 81 | 9 | 405.00 |
Mathematical reference: US National Institute of Standards and Technology (NIST).
Pyramid Frustum Volume Calculator: frequently asked questions
What is a pyramid frustum?
A frustum is the portion of a pyramid or cone left after the top is cut off by a plane parallel to the base. It has two parallel faces of different sizes joined by sloping sides. Everyday examples include a bucket, a lampshade, a tapered planter and many embankment and hopper shapes.
Does this work for cones too?
Yes. The same formula works for a conical frustum because it depends only on the two parallel cross-sectional areas and the height, not on the shape of the cross-section. For a cone, supply the circular areas of the two ends, each equal to pi times the radius squared.
What does the square-root term represent?
It is the geometric mean of the two face areas, the square root of their product. It accounts for the gradual taper between the large and small faces. Without it the formula would over- or under-estimate the volume, because the cross-sectional area does not change linearly with height.
What units does the result use?
The volume comes out in cubic units of whatever length unit you used for the areas and height. If the areas are in square meters and the height is in meters, the volume is in cubic meters. Keep all inputs in consistent units.
What is the frustum volume formula?
The volume equals one third the height times the sum of the two face areas plus the square root of their product: V = (h/3)(A1 + A2 + sqrt(A1 A2)). For areas 100 and 36 with height 9 that is 3 times 196, which is 588.
Official sources
- Mathematical functions and computational reference data (Digital Library of Mathematical Functions): US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.