Pyramid calculator
Enter the base length, width and height of a right rectangular pyramid to get its volume, base area, lateral surface area and total surface area. It is useful for geometry coursework, construction, packaging and any project involving a four-sided pyramid.
How it works
With base length l, width w and vertical height h, the tool computes:
base area = l × w
volume = (1/3) × l × w × h
slant_l = √(h² + (w/2)²) slant_w = √(h² + (l/2)²)
lateral = l × slant_l + w × slant_w
total = lateral + base areaThe two slant heights reach the midpoints of opposite base edges, which is why each uses half of the other dimension. The apex is assumed to sit directly above the centre of the base.
Example
A pyramid with l = 6, w = 6, h = 9:
- Base area: 6 × 6 = 36
- Volume: (1/3) × 36 × 9 = 108
- Slant heights: √(9² + 3²) = √90 ≈ 9.487 (both, since l = w)
- Lateral: 6 × 9.487 + 6 × 9.487 ≈ 113.84
- Total surface area: 113.84 + 36 ≈ 149.84
| Quantity | Formula | Value |
|---|---|---|
| Base area | l × w | 36 |
| Volume | ⅓·l·w·h | 108 |
| Lateral area | l·slant_l + w·slant_w | 113.84 |
| Total area | lateral + base | 149.84 |
Understanding slant height
The slant height is not the same as the true height. The true (vertical) height h runs straight up from the base to the apex. The slant height runs along the face of the pyramid from the apex down to the midpoint of a base edge — it is the hypotenuse of a right triangle formed by the apex, the base centre, and the midpoint of the edge.
A rectangular base has two distinct slant heights because the two pairs of faces have different widths. One pair of triangular faces is as wide as the base length l, and their slant height uses w/2 in the Pythagorean formula. The other pair is as wide as w, and uses l/2. For a square pyramid, both slant heights are equal.
Practical applications
Packaging design — pyramid-shaped gift boxes, display stands, or specialty containers need surface area to estimate material usage and volume to ensure fit.
Architecture and scale models — a hipped roof on a rectangular building is essentially a pyramid. Calculating its lateral area gives the roofing material needed, and its volume helps with structural load estimates.
Geometry coursework — the one-third relationship between a pyramid and a prism of the same base and height is a classic result. A prism with the same base and height as this pyramid would have volume l × w × h; the pyramid is exactly one-third of that.
Sand and gravel stockpiles — material piled into a cone or pyramid shape can be volume-estimated using this formula, which helps with logistics and load planning.
Square pyramid as a special case
A square pyramid is simply a rectangular pyramid with equal base dimensions (l = w). Enter the same value for both and the calculator works identically. The two slant heights become equal, and the lateral area simplifies to 2 × l × slant_height.
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