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 center to apex. Same 1/3 factor as every other pyramid — pyramids are 1/3 the volume of their bounding prism.

A rectangular pyramid has a rectangular base (not necessarily square) and four triangular sides meeting at a single point above. Two of the triangular faces are usually congruent to each other, and the other two are also congruent to each other (but different from the first pair) — unless the base is square, in which case all four faces match.

Worked example — hipped roof on a non-square house: A house with a 30 ft × 50 ft footprint has a hipped roof that meets at a single ridge directly over the center. Wait — that’s only true if the house is square. For a rectangular base, a true hipped pyramid would have all four faces ending at one point, which means the apex is the geometric center and the long-axis hips are steeper than the short-axis hips.

In practice, rectangular hipped roofs use a horizontal ridge along the longer axis rather than a single apex point — that’s the “true hip” geometry. Pure pyramid hips are only seen on square or near-square buildings.

For a 12 ft × 12 ft true pyramidal hip with a 6 ft rise: V = (1/3) × 144 × 6 = 288 ft³ of attic space.

Where rectangular pyramids show up:

• Pyramidal hopper bottoms. Many industrial bins have rectangular cross-sections with pyramidal hoppers at the bottom. The volume of the hopper part is rectangular-pyramid volume.
• Square or near-square pavilion roofs. Garden pavilions, gazebos, small park structures.
• Tetrapack pyramid milk cartons. Old-style Tetra Classic packs from the 1950s-70s were tetrahedral, not rectangular — but the concept is similar.
• Cake molds. Pyramidal silicone cake pans for novelty desserts.
• Cement form work. Pyramidal column-base forms for footings.

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”:

For a rectangular pyramid, even more height confusion than for the square version:

• h (perpendicular height): straight down from apex to base plane. Volume formula uses this.
• slant heights for the two pairs of faces: these are different — one for the long-base triangles, one for the short-base triangles.
• edge length (apex to a base corner): longest of all.

Always use perpendicular h for volume.

Sanity check:

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