The Luxor Hotel in Las Vegas is a square pyramid 350 feet tall on a 620-foot base — and its volume is one-third the base area times the height: (620 × 620 × 350) / 3 ≈ 44.8 million cubic feet of hotel. Every pyramid, from ancient Giza to a tabletop paperweight, follows V = (base area × height) / 3, because three pyramids exactly fill the prism with the same base and height. The base here is a rectangle, so its area is length × width first. That one-third factor is the whole story of pyramid volume, and forgetting it is the single most common pyramid error: the un-divided product is the prism's volume, three times too big.

The formula, and why one-third is exact

A pyramid tapers from a full base to a single point, so its average cross-section over its height is exactly one-third of the base, not roughly. Slice the pyramid into infinitely thin horizontal layers: the bottom layer has the full base area, the top layer has none, and the area shrinks quadratically in between. Summing those layers gives exactly one-third of base times height, which is why the factor is 3 and not an approximation like 2.9 or 3.1. The same one-third rule governs cones, triangular pyramids, and any pointed solid with straight sides, and it is the reason the pyramid and the cone share a formula shape: both are one-third of their enclosing prism or cylinder. Whenever a pyramid volume looks too big, check the division first; a missing divide-by-3 triples the answer and triples the material order that follows from it.

Worked example: a pyramid planter

A backyard pyramid planter 4 feet by 4 feet and 3 feet tall shows 16 cubic feet of soil capacity: base area 16 square feet times height 3 is 48, divided by 3 is 16. That 16 cubic feet is about 0.59 cubic yards, and at 2 cubic feet per bag it takes 8 bags to fill, a number a gardener can buy in one trip. A bigger planter, 5 feet by 5 feet and 4 feet tall, holds 25 times 4 divided by 3, about 33.33 cubic feet, which is 33.33 divided by 27, about 1.23 cubic yards. Bagged, that is 33.33 divided by 2, about 16.67 bags, so 17 bags cover it. Ordering by the yard instead, round 1.23 up to 1.5 yards: soil settles and spills, and a short load is worse than a small surplus. The planter's sloped sides mean the top rows of plants sit in less soil than the bottom rows, so deep-rooted plants belong near the center where the soil is deepest.

Worked example: a pyramid tent

A camping pyramid tent 7 × 7 feet with a 5-foot peak encloses about 81.7 cubic feet of sleeping space: 49 times 5 is 245, divided by 3 is 81.67. That number decides between cozy and cramped: two sleepers get about 40 cubic feet each, roughly a small closet apiece, which is honest single-wall tent math. Compare it with a prism tent of the same footprint and peak height, which would enclose 245 cubic feet, three times as much, and the trade-off is obvious: the pyramid sheds wind and snow far better, but it charges for that strength in headroom, since only the center pole reaches the full 5 feet. A 9 × 9 foot pyramid tent with a 6-foot peak encloses 81 times 6 divided by 3, exactly 162 cubic feet, double the smaller tent, which shows how quickly footprint growth pays: adding 2 feet to each side nearly doubles the volume.

Worked example: a concrete pyramid footing

A square pyramid footing 6 feet by 6 feet and 4 feet tall needs 36 times 4 divided by 3, exactly 48 cubic feet of concrete. Suppliers sell by the yard, so divide by 27: 48 divided by 27 is about 1.78 cubic yards, and ordering 2 yards covers waste and spillage. Bagged, an 80-pound bag yields about 0.6 cubic feet, so 48 divided by 0.6 is exactly 80 bags, a full pallet-scale job that argues for a ready-mix truck instead. The pyramid shape is not decorative here: the wide base spreads the load over 36 square feet while the narrow top meets the post, so the footing resists overturning far better than a straight 6-foot cube would, and it uses one-third of that cube's concrete to do it. Footings, hoppers, and pyramid roofs all exploit the same bargain: full base strength at one-third the material.

The Great Pyramid, by the same arithmetic

The Great Pyramid of Giza runs about 756 feet along each base side and 481 feet tall, and the same three-step arithmetic sizes it. Base area is 756 times 756, which is 571,536 square feet. Times height 481 is 274,908,816, and divided by 3 is 91,636,272 cubic feet of pyramid. In cubic yards that is 91,636,272 divided by 27, about 3,393,936 yards, a figure that makes the ancient logistics vivid: at one cubic yard per cartload, the pyramid is roughly 3.4 million cartloads, before mortar and waste. The modern Luxor comparison sharpens it: the Luxor's 44.8 million cubic feet is less than half of Giza's volume, despite looking enormous on the Strip, because Giza is both wider and taller. Scaling is cubic and unforgiving: a pyramid twice as wide and twice as tall holds eight times the volume, which is why small measurement errors on big pyramids explode into huge volume errors.

Triangular pyramids use the same divide-by-three

Only the base area changes when the base is a triangle instead of a rectangle. A triangular pyramid whose base triangle measures 10 feet by 8 feet has a base area of half of 10 times 8, which is 40 square feet. With a height of 9 feet, the volume is 40 times 9 divided by 3, exactly 120 cubic feet. The divide-by-3 never changes, and neither does the vertical-height rule: the height is the perpendicular distance from the base plane to the apex, not the slanted face height. A tetrahedron, the triangular pyramid with all edges equal, is the special case students meet first: for edge 6, the base area is about 15.59 square feet and the height is about 4.90 feet, giving a volume of about 25.46 cubic feet, but the rectangular-base version on this calculator is the one that sizes planters, footings, and tents. If a triangular base is what you have, compute its area first, then the same three steps carry you home.

Running the formula backward

Often the volume is known and a dimension is missing, and the formula rearranges cleanly. A pyramid holds 108 cubic feet on a 6-by-6-foot base: base area 36, so the height is 3 times 108 divided by 36, which is 324 divided by 36, exactly 9 feet. A pyramid roof must enclose 500 cubic feet with a 10-foot height: the base area must be 3 times 500 divided by 10, which is 150 square feet, so a 15-by-10-foot base or a 12.25-foot square base both work. The backward form is also the check on every forward answer: multiply base area times height, divide by 3, and compare with the claimed volume. If they disagree, the usual suspects are a slant height used as vertical height, or a base area computed by adding the sides instead of multiplying them.

Slant height and the faces

Volume is only half of pyramid geometry; the faces matter whenever material covers the outside. For a square pyramid with a base side of 8 feet and a vertical height of 12 feet, the slant height is the hypotenuse of the height and half the base side: the square root of 144 plus 16, the square root of 160, about 12.65 feet. Each triangular face has area half of base times slant height, half of 8 times 12.65, about 50.59 square feet, and four faces give a lateral area of about 202.39 square feet. Add the 64-square-foot base and the total surface is about 266.39 square feet, the figure a roofer or painter needs. At 400 square feet per gallon, one coat takes 266.39 divided by 400, about 0.67 gallons, so one gallon covers it with room for a second coat on the windward faces. Never price a pyramid roof from the volume: volume buys fill, surface buys skin, and confusing the two is an expensive mistake.

Scaling: double every side, eight times the volume

Pyramid volume scales with the cube of the linear scale, and the numbers prove it fast. A pyramid 3 feet by 3 feet and 2 feet tall holds 9 times 2 divided by 3, exactly 6 cubic feet. Double every dimension to 6 by 6 by 4 feet and the volume is 36 times 4 divided by 3, exactly 48 cubic feet, eight times as much. Double only the height, to 3 by 3 by 4 feet, and the volume merely doubles to 12 cubic feet. Double only the base sides, to 6 by 6 by 2 feet, and the volume quadruples to 24 cubic feet. This ladder is the estimator's friend: a planter that looks "about twice as big" is actually eight times the soil if all three dimensions doubled, and the bag count must follow the cube, not the eye. The same law governs the Luxor-to-Giza comparison above and every scale model in an architect's office.

Unit conversions that finish the job

Suppliers rarely sell in the unit you measured in, so the conversions belong with the formula. Cubic feet to cubic yards: divide by 27, since a yard is 3 feet and 3 cubed is 27. The 33.33-cubic-foot planter is 1.23 yards, and the 48-cubic-foot footing is 1.78 yards. Cubic inches to US gallons: divide by 231, the wine-gallon definition. Cubic centimeters to liters: divide by 1,000. Cubic feet to US gallons: multiply by 7.48052, so the 33.33-cubic-foot planter holds about 249.35 gallons of water if it were watertight. The conversion that bites most often is inches to feet before multiplying: a planter measured as 60 by 60 by 48 inches is 5 by 5 by 4 feet, and multiplying the inch figures first gives 172,800 cubic inches, which divided by 1,728 is 100 cubic feet, not 33.33 — the inch version is a different planter unless the division by 1,728 happens. Convert first, multiply second, and the units stay honest.

How to use this calculator

Type the Base Length (l), Base Width (w), and Height (h) into their boxes and read the Volume output — the calculator multiplies all three and divides by 3 in one step. A backyard pyramid planter 4 feet by 4 feet and 3 feet tall shows 16 cubic feet of soil capacity. A camping pyramid tent 7 × 7 feet with a 5-foot peak encloses about 81.7 cubic feet of sleeping space, the number that decides between cozy and cramped. For soil or concrete, add 5–10% over the exact volume for settling and spillage. Compare with a prism of the same base and height: the pyramid always holds exactly one-third, so a prism answer that matches your pyramid answer means the division was skipped.

Five mistakes that cost real material

The first mistake is forgetting the divide-by-3, which triples the order. The second is using the slant height as the vertical height; the slant is always longer, so the volume inflates. The third is computing the base area by adding the sides instead of multiplying them, which collapses the area and the volume together. The fourth is mixing units, inches with feet, which corrupts the product by a factor of 12 or 1,728 depending on how many dimensions are mixed. The fifth is pricing surface work from the volume or fill work from the surface: volume buys soil and concrete, surface buys paint and roofing, and the two numbers for the same pyramid are never interchangeable. Run the one-third check (volume times 3 must equal base area times height) after every calculation.

Related calculators on this site

If the pyramid is one shape in a bigger job, three companion tools carry the next step. The volume of cone calculator handles the round cousin, where the base area is pi times radius squared and the same divide-by-3 applies. The volume of cylinder calculator sizes the enclosing prism-like shape, the three-times reference for any pyramid or cone check. The volume of cuboid calculator prices the rectangular boxes and planters that pyramids sit beside, and the triangular prism calculator covers the wedge shapes that roofers meet between pyramids and boxes. Each one uses the same discipline: base area first, vertical height second, and the right divisor last.

A classroom note on why three pyramids fill a prism

Students often ask why the factor is exactly 3 rather than a rounded approximation, and there is a hands-on answer: three identical square pyramids with base side s and height s assemble into a cube of side s, with no gaps and no overlap. Pour sand into a pyramid, tip it into its prism three times, and the prism fills exactly. The calculus version says the same thing: cross-sections shrink quadratically from base to apex, and integrating that quadratic from base to tip gives exactly one-third. That is also why the rule survives every base shape: triangle, square, hexagon, or circle, the taper is the same and the third is the same. A pyramid is a prism that learned to point, and the point costs it two-thirds of its volume.

With the pre-filled defaults — Base Length (l) 4, Base Width (w) 3, Height (h) 5 — the Volume of Pyramid Calculator returns Volume 20.0000 m³. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.