Volume of a Rectangular Pyramid Calculator
Enter the base length, base width, and vertical height in any one unit to get the pyramid's volume, base area, the matching prism volume, and both slant heights.
Example: with Base length (l) 8 · Base width (w) 6 · Vertical height (h) 9 → Volume: 144 cubic units.
- Base area (l × w)48 square units
- Matching box (prism) volume432 cubic units — the pyramid is exactly 1/3 of it
- Slant heights9.487 units (faces on the length sides), 9.849 units (faces on the width sides)
Computed by the calculator below using its default values. Change any input to see your own numbers.
V = lwh/3 — the same one-third rule as every pyramid and cone.
One-third of the box
A rectangular pyramid holds exactly one-third of the rectangular prism sharing its base and height: V = (l × w × h)/3. The default 8 × 6 base with height 9 gives a 48-square-unit floor, a 432-cubic-unit box, and a 144-cubic-unit pyramid. The 1/3 factor is exact, not rounded — the same rule that makes a cone one-third of its cylinder.
Remarkably, the volume does not care where the apex sits. Slide the apex sideways at the same height and every horizontal slice keeps its area (Cavalieri's principle), so V = lwh/3 holds even for a leaning, oblique pyramid.
Two slant heights, not one
Unlike a square pyramid, a rectangular pyramid has two different slant heights: the faces rising from the long edges climb √(h² + (w/2)²), while the faces on the short edges climb √(h² + (l/2)²). Volume never uses either — it wants the vertical height — but you will need both the moment the problem turns to surface area.
How it’s calculated
V = (l × w × h)/3 — one-third of the l × w × h box. Slant heights: √(h² + (w/2)²) for the faces on the length sides and √(h² + (l/2)²) for the faces on the width sides. Rounded to 3 decimals.
The volume holds even with an off-center apex (Cavalieri's principle), but the two slant heights assume a right pyramid with the apex over the base center.
Worked examples
| l × w × h | Volume (lwh/3) |
|---|---|
| 8 × 6 × 9 | 144 |
| 10 × 4 × 6 | 80 |
| 5 × 5 × 12 | 100 |
| 12 × 9 × 10 | 360 |
Computed with V = lwh/3.
Common mistakes
- Using a slant height as h — volume needs the vertical height from base plane to apex.
- Dividing by 2 instead of 3 (or forgetting to divide): a pyramid is exactly one-third of its box.
- Mixing up the two slant heights when moving on to surface area — each pair of faces has its own.
- Multiplying l, w, and h in different units and reporting a meaningless product.
Frequently asked questions
What is the volume formula for a rectangular pyramid?
V = (l × w × h)/3: base length times base width times vertical height, divided by 3. With an 8 × 6 base and height 9 that is 432/3 = 144 cubic units.
Why divide by 3?
Any pyramid or cone fills exactly one-third of the prism or cylinder with the same base and height — provable by dissection and by integration. It is a general rule, not specific to rectangles.
Does the apex have to be over the center of the base?
Not for volume. By Cavalieri's principle, only the perpendicular height matters, so an oblique pyramid with the same base and height holds the same amount. The slant heights shown do assume a centered apex.
What is the difference between the height and the slant heights?
The height h is vertical, base to apex. Each slant height runs down the middle of a triangular face: √(h² + (w/2)²) on the long-edge faces and √(h² + (l/2)²) on the short-edge faces. Both exceed h.