A pyramid frustum, or truncated pyramid, is what remains when a cut parallel to the base removes the top of a pyramid. The two ends are similar regular polygons, and the sides are trapezoids. Square versions are everywhere: grain and sand hoppers, sloped wooden planters, tapered concrete footings and the collection bins under chutes. This calculator handles any regular base from 3 to 100 sides. The default, n = 4, is the common square frustum.
How to use the pyramid frustum calculator
- Set Sides of the base (n): 4 for a square frustum, 3 for triangular, 6 for hexagonal.
- In I know the, choose Both edges and the height, Both edges and the slant height or Both edges and the volume.
- Enter the Bottom edge (a) and Top edge (b), one side of each polygon, then the Height (h), the Slant height (s) measured along a side face, or the Volume (V) in cubic units. Pick the Units.
- Read the volume, or the height in volume mode. The list adds the base shape, capacity in liters and US gallons, total surface area, lateral area, bottom and top base areas (A₁ and A₂), slant height and lateral edge.
Pyramid frustum formulas
For any frustum whose ends have areas A₁ and A₂:
For a square base this becomes:
For other polygons, the slant height uses the difference between the two apothems (center to middle of a side) in place of (a − b)/2, and the lateral edge uses the two circumradii (center to corner).
Worked example: a sloped wooden planter
A square planter is 14 in across the bottom and 20 in across the top inside, with 16 in of vertical depth. Keep n = 4, choose Both edges and the height, and enter a = 14, b = 20 and h = 16 in inches.
- Volume: ⅓ × 16 × (196 + 280 + 400) = 4,672 in³, about 2.70 ft³ of soil
- Capacity: about 76.56 L or 20.23 US gal
- Slant height: √(16² + 3²) ≈ 16.28 in, the height of each side panel measured along its face
- Lateral area: 4 × (14 + 20) × 16.28 ÷ 2 ≈ 1,106.96 in², or about 7.69 ft² of liner for the four walls
The same frustum with a hexagonal base (n = 6, edges of 10 and 6 in, 12 in tall) holds 2,036.89 in³, about 8.82 US gal.
Tapered concrete footings
A pier footing 2 ft square at the bottom, 1 ft square at the top and 1 ft tall holds ⅓ × 1 × (4 + 2 + 1) ≈ 2.33 ft³, or about 0.086 yd³ after dividing by 27. Eight of them need roughly 0.69 yd³ before any allowance for waste; the concrete calculator can turn that into bags.
Why not just average the two ends?
Multiplying the height by the average of the end areas overestimates, and squaring the average edge underestimates. For the planter, (196 + 400) ÷ 2 × 16 = 4,768 in³ is about 2% high, while 17² × 16 = 4,624 in³ is about 1% low. The exact formula adds the geometric mean of the two end areas, √(A₁A₂), which for a square base is the middle term ab.
Surveyors estimating earthwork use the prismoidal formula, V = h/6 × (A1 + 4M + A2), where M is the area of the cross-section halfway up. For any frustum it is exact: the planter’s mid-section is 17 in square, and 16/6 × (196 + 4 × 289 + 400) = 4,672 in³.
The oldest volume problem on record
The Moscow Mathematical Papyrus, written in Egypt around 1850 BC, contains a worked frustum problem: a square frustum 6 units high with a bottom edge of 4 and a top edge of 2. Enter those numbers here and you get 56 cubic units, the same answer the scribe reached by adding 4², 4 × 2 and 2², then multiplying by a third of the height. A rigorous Greek proof of the one-third rule for pyramids did not appear until Eudoxus and Euclid, well over a thousand years later.
For a pyramid that runs all the way to a point, use the square pyramid calculator. Round hoppers and buckets belong in the conical frustum calculator, and the regular polygon calculator gives the end areas on their own.
Frequently asked questions
What is the volume formula for a square pyramid frustum?
V = ⅓h(a² + ab + b²), where a and b are the bottom and top edges and h is the vertical height. A frustum with edges of 8 and 4 and a height of 6 has V = ⅓ × 6 × (64 + 32 + 16) = 224 cubic units.
How does the calculator handle a hexagonal or triangular base?
Set Sides of the base (n) to the number of sides. The calculator finds each end's area from its edge with the regular polygon formula A = na² ÷ (4 tan(180°/n)), then applies V = ⅓h(A₁ + A₂ + √(A₁A₂)). Both ends must be regular polygons with the same number of sides.
What is the slant height of a pyramid frustum?
It is the height of one trapezoid side face, measured along the face from the middle of the bottom edge to the middle of the top edge. For a square frustum it equals √(h² + ((a − b)/2)²). It is the height to cut each side panel of a sloped planter or hopper, not the vertical height.
What did the Moscow Mathematical Papyrus calculate?
Its Problem 14 asks for the volume of a square frustum 6 units high with a bottom edge of 4 and a top edge of 2. The scribe squared 4, multiplied 4 by 2, squared 2, added them to get 28 and multiplied by a third of 6, giving 56, which is exactly the modern formula.
What if the top edge is zero or equal to the bottom edge?
A top edge of 0 gives a complete pyramid, and equal edges give a straight-sided prism. The calculator still returns a result in both cases and adds a note saying which shape you have.