Pyramid Frustum Volume Calculator
From two square sides and a height, get the volume, the lateral surface, and the total surface area — the numbers that describe any square pyramid with its tip cut off.
Three inputs, three answers
Enter the bottom side, the top side, and the height and the calculator returns the volume ((h/3)(a² + ab + b²)), the lateral surface, and the total surface at once.
Keep units consistent
The two sides and the height are unit-agnostic — your answers come back in the same unit (squared for the surfaces, cubed for the volume), so don't mix centimetres with inches.
What is a pyramid frustum?
A square pyramid with its tip sliced off
A frustum of a square pyramid is what you get when you take a square pyramid and slice off the top with a cut parallel to the base. You are left with a flat-topped solid: a large square at the bottom (side a), a smaller square at the top (side b), and four sloping faces joining them. It is the shape of a lampshade, a planter, a cake tier, a hopper, or a step on a Mesoamerican temple. Knowing the two square sides and the vertical height fixes every other measurement, because the faces all slope at the same angle.
Enter the bottom side, top side, and height in any length unit to get the volume, lateral surface, and total surface instantly.
Three short formulas, all built from the bottom side a, the top side b, and the height h.
volume = (h/3) × (a² + a·b + b²)The slant height is the diagonal height of a sloping face; by the Pythagorean theorem it is √(h² + ((a − b)/2)²), where (a − b)/2 is how far each side steps inward. The lateral surface — the four trapezoid faces — is 2 × (a + b) × slant height. The total surface adds the bottom square (a²) and the top square (b²) to that.
Suppose you have a frustum with a bottom side of 6, a top side of 4, and a height of 9.
Slant height
√(9² + ((6 − 4)/2)²) = √(81 + 1) = √82 ≈ 9.055385 — the diagonal of a sloping face.
Lateral surface
2 × (6 + 4) × 9.055385 ≈ 181.107703 square units — the four slanted faces.
Volume
(9/3) × (36 + 24 + 16) = 3 × 76 = 228 cubic units — the space inside.
The three outputs answer three different practical questions. The volume (228 cubic units for a = 6, b = 4, h = 9) is how much the frustum holds — the soil in a tapered planter, the material in a hopper, the batter in a cake tier. Notice that the formula sits neatly between two simpler ones: set the top side equal to the bottom side and it collapses to a box (a² × h), set the top side to zero and it collapses to a full pyramid ((1/3)a²h). The slant height (about 9.055385 here) is the diagonal you would measure down a sloping face, always a little longer than the vertical height — it is the number you need when cutting the side panels for a lampshade or a planter. The lateral surface (about 181.107703 square units) is just those four faces, the material you would wrap or skin; the total surface (about 233.107703) adds the bottom and top squares for the full outside area, useful when you need to paint or cover the whole solid. Reach for lateral when the top and bottom are open, total when they are closed.
The formulas are exact, but a couple of practical points are worth keeping in mind.
Right square frustums and consistent units
These formulas describe a right frustum of a square pyramid — both faces are squares, centred one above the other, with the height measured straight up. A rectangular frustum (different-sized rectangles), an oblique frustum (top offset to one side), or a frustum with non-square faces will differ from the computed value. The two sides and the height are also unit-agnostic, so the answers are only meaningful if you keep one unit throughout: sides and a height in centimetres give a volume in cubic centimetres and surfaces in square centimetres, never a mix.