Quick answer: the volume of a rectangular pyramid is V = (1/3) × l × w × h — one-third of the enclosing box. Base 3 × 4, height 6: (1/3) × 72 = 24 cubic units. Square base? The square pyramid calculator needs one fewer measurement.
The rectangular pyramid formula, step by step
where l, w = base rectangle sides · h = vertical height (base plane to apex)
Worked example — a display case topper 3 ft long, 4 ft wide, 6 ft tall:
- Base area: 3 × 4 = 12 ft²
- Times height: 12 × 6 = 72
- One-third: 24 ft³
Because the base has two different side lengths, this pyramid has two face slant heights — one over the long sides, one over the short. If you measured a slant instead of the vertical height, both conversions are in the FAQ below, and getting the same h from each is a free accuracy check.
Where rectangular pyramid volume shows up in real life
Skylight wells
A pyramid skylight over a 4 × 6 ft opening, peaking 2 ft above the roof plane, encloses (1/3) × 24 × 2 = 16 ft³ — the added air a room's HVAC load calculation has to absorb.
Grain and pellet hoppers
Rectangular bins feed augers through inverted rectangular pyramids. A 6 × 4 ft bin bottom tapering over 3 ft: (1/3) × 24 × 3 = 24 ft³ of live storage in the taper. Bins that narrow to a slot rather than a point are wedges — closer to a triangular prism.
Hip roof ends
The triangular end sections of a hip roof are half-pyramids. A hip end over a 24-ft-wide house, extending 8 ft in plan and rising 6 ft, contributes (1/3) × 24 × 8 × 6 ÷ 2... careful — it's easier to model the full attic as prism + two pyramid ends: each end is (1/3) × 24 × 8 × 6 = 384 ft³.
Common mistakes & pro tips
- Slant height as vertical height. With two different slants, it's doubly easy to slip. Vertical height is always the shortest of the three.
- Halving l and w. The formula uses full base sides, not half-sides (those belong only inside slant conversions).
- Forgetting the ÷3. lwh alone is the box over the same base.
- Pro tip — frustum tops. A pyramid cut flat at the top (planter, lampshade base) = big pyramid − small pyramid. Run this calculator twice and subtract.
Frequently asked questions
What is the formula for the volume of a rectangular pyramid?
V = (1/3) × l × w × h — one-third of the box with the same base and height. A pyramid over a 3 × 4 base rising 6 units holds (1/3) × 3 × 4 × 6 = 24 cubic units.
How is a rectangular pyramid different from a square pyramid?
Only the base: a rectangle with unequal sides instead of a square. The volume rule is identical — one-third of base area times height — but a rectangular pyramid has two different slant heights, one per pair of faces, which matters if you're converting from a slant measurement.
Which slant height do I convert from?
A rectangular pyramid has two: across the width (h = √(s₁² − (w/2)²)) and across the length (h = √(s₂² − (l/2)²)). Both must give the same vertical height — computing it both ways is a good check on your measurements.
Does it matter if the apex isn't centered?
Not for volume. By Cavalieri's principle, any pyramid with the same base rectangle and the same apex height holds (1/3)lwh, even if the apex leans far off-center — only the slant lengths change.