V = (l × w × h) / 3, where l and w are the length and width of the rectangular base and h is the vertical height from the base to the apex. A pyramid fills exactly one-third of the rectangular box (prism) with the same base and height — that is where the division by 3 comes from.

A (non-square) rectangular pyramid has two different slant heights, one for each pair of opposite triangular faces. The slant height of the faces built on the base edge l is s_l = √(h² + (w/2)²), measured from the apex perpendicular down to that edge. The slant height of the faces built on the base edge w is s_w = √(h² + (l/2)²). For a base of 2 × 4 with height 3: s_l = √(3² + 2²) = √13 ≈ 3.6056 and s_w = √(3² + 1²) = √10 ≈ 3.1623.

Add the areas of the four triangular faces: lateral area = l·s_l + w·s_w, where s_l and s_w are the two slant heights. (Each face must pair with its own slant height — the face on base edge l uses s_l.) For a base of 2 × 4 with height 3, the lateral area is 19.8602 square units.

Total surface area = lateral area + base area = (l·s_l + w·s_w) + l·w. The base counts once; the apex contributes no extra area. For a base of 2 × 4 with height 3, the total surface area is 27.8602 square units.

Volume: V = 2 × 4 × 3 / 3 = 8 cubic units. Slant heights: s_l = √(3² + 2²) = √13 ≈ 3.6056, s_w = √(3² + 1²) = √10 ≈ 3.1623. Lateral area = 2 × s_l + 4 × s_w ≈ 19.8602 square units. Total surface area = lateral + base = 19.8602 + 8 ≈ 27.8602 square units.

The slant height drops from the apex perpendicular to the midpoint of a base edge (two of them: 3.6056 and 3.1623 for a 2 × 4 base with height 3). The lateral edge runs from the apex to a base corner: e = √(h² + (l/2)² + (w/2)²), which is ≈ 3.7417 here — always the longest of the three.

Then it is a square pyramid: the two slant heights merge into one, s = √(h² + (a/2)²) where a is the base side. Volume V = a²h/3, lateral area = 2·a·s, total = 2as + a². Example: a = 6, h = 4 gives V = 48, s = √(16 + 9) = 5, lateral = 60, total = 96.

No — this calculator is for a full rectangular pyramid whose apex is intact. A frustum is a pyramid with its top cut off by a plane parallel to the base, and it needs a different formula: V = h/3 × (A1 + A2 + √(A1·A2)). For that shape, use our pyramid frustum calculator.

Any consistent length unit works — meters, centimeters, inches, feet, whatever you measure in. Enter l, w, and h all in the same unit; the volume comes out in cubic units (unit³) and all areas in square units (unit²). The unit selector above only changes the labels, never the math.

Yes — rearrange the volume formula: h = 3V / (l×w). For example, a pyramid with volume 48 and a 6 × 6 base has height h = 3 × 48 / 36 = 144 / 36 = 4.