Rectangular Pyramid Volume Calculator

V = (1/3) × l × w × h

Where l is the base length, w is the base width, and h is the perpendicular height from base centre to apex. The 1/3 is universal: every pyramid holds a third of the prism standing on the same base at the same height, whatever the base shape.

A rectangular pyramid has a rectangular base and four triangular sides meeting at one point above it. Those four faces come in two congruent pairs, and the pairs differ from each other unless the base happens to be square.

Worked example: attic space under a pyramidal hip

A 12 ft × 12 ft pavilion with a 6 ft rise to the apex: V = (1/3) × 144 × 6 = 288 ft³ of enclosed roof space.

Compare that with the box beneath it. A 12 × 12 × 6 ft room holds 864 ft³, so the roof adds a third as much again. That ratio is fixed, which makes it a quick mental check: whatever the prism holds, the pyramid on top holds a third.

A note on real hipped roofs. A true pyramid hip needs a square or near-square footprint, because all four faces have to reach the same apex. Stretch the base to 30 × 50 ft and the two faces on the short ends have to climb across 25 ft of plan to get there while the long faces climb only 15, so they end up at wildly different pitches and the roof looks wrong. Long buildings use a horizontal ridge with hipped ends instead, which is a different shape and needs a different calculation.

Where rectangular pyramids show up:

• Pyramidal hopper bottoms. Industrial bins with rectangular cross-sections taper to a pyramidal hopper, and that taper is exactly this shape.
• Square and near-square pavilion roofs. Garden pavilions, gazebos, small park structures.
• Footing forms. Pyramidal or tapered column bases in concrete work.
• Cake and chocolate moulds. Pyramidal silicone pans for novelty desserts.
• Excavation spoil and stockpiles, at least approximately, when a heap is dumped against two walls.

Comparing to a rectangular prism (box):

A rectangular box with the same l, w, h has volume l × w × h. The pyramid is 1/3 of that. So if you’re sizing a hopper that needs to hold 30 ft³ of grain before emptying, and the inside dimensions of the hopper are 5 ft × 5 ft at the top, you need a 3.6 ft tall pyramidal hopper: 30 = (1/3) × 25 × h → h = 3.6 ft.

Two different “heights”:

There are four different lengths from the apex, and only one of them belongs in the volume formula:

• h, the perpendicular height. Straight down from the apex to the base plane. This is the one volume uses.
• The slant to the middle of the long base edge, √(h² + (w/2)²).
• The slant to the middle of the short base edge, √(h² + (l/2)²). Different from the one above.
• The hip, apex to a base corner, √(h² + (l/2)² + (w/2)²). Longest of the four.

Volume takes h. Surface area takes the two slants. Rafters get cut to the hip. Reaching for the wrong one is the most common error on this shape, and the hip in particular is the one people measure and then use everywhere.

Sanity check:

• l = w (square base): collapses to the square pyramid formula. ✓
• l, w, h all = 1: V = 1/3. ✓