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Pyramid Volume Calculator

Range: 0 cm² – 1,000,000,000,000,000,000 cm²

Range: 0 cm – 1,000,000,000 cm

Result

48.00 cm³

Volume

A pyramid volume calculator takes the area of the base and the height of the pyramid and returns the volume it encloses: base area times height over three. It is the shortest formula on this site and the one with the most surprising constant in it. Three identical pyramids fit exactly into a prism of the same base and height, no matter what polygon the base is — a square, a triangle, a hexagon — which is where the division by three comes from, and it is why the volume of a pyramid is a third of what a box of the same footprint and height would hold. That relationship is worth holding onto, because it turns a hard question into an easy one: if you can work out the volume of the prism, you already know the pyramid's. The page asks for the base as an area rather than as the lengths of its edges, and that is deliberate. A pyramid's base can be any polygon at all — a square, a triangle, a hexagonal lamp shade, a five-sided tent base — and the formula only ever uses the base through its area, so asking for two edges would restrict the page to rectangular bases without making the arithmetic any shorter. It also means the page cannot give a slant height, which is the distance from the apex down the middle of a face to the middle of an edge: that distance depends on the shape of the base and not only on its area, and a long thin base and a compact one with the same area have quite different slant heights. Measuring the base's area in the first place is usually a separate step, and the area calculators for rectangles, squares, triangles and circles are all on this site; this page takes the result of that step and does the three-dimensional part. Two examples fix the feel of the numbers. A pyramid with a base of 36 square centimetres and a height of four has a volume of 48 cubic centimetres, and a prism with the same base and height holds three times that, 144. And the largest pyramid ever built, the Great Pyramid at Giza, has a base of about 52900 square metres — 230 metres on a side — and a height of about 146.6 metres, which gives roughly 2.6 million cubic metres of stone. That is the scale at which the division by three stops being trivia and starts being the difference between a pile of rock and a mountain. The table below runs from a base of one square centimetre up to that pyramid, and the last row is the page's smallest legitimate answer: a pyramid with no base or no height has no volume, which is a result rather than an error.

Pyramids from a base of one square centimetre to the Great Pyramid at Giza

Base area (cm²)Height (cm)Volume (cm³)
36448
110.33
1009300
48348
10012400
25001210000
529000000146602585046666666.67
000

Seven pyramids and an empty one. The first row is the size the page loads with. The third row is the one to test the arithmetic against, because every number in it is exact: a base of 100 square centimetres with a height of 9 has a volume of 300, and a box with the same base and height holds 900 — the prism is always exactly three times the pyramid, on every row of this table. The second row is where the division by three shows its hand: a base of 1 and a height of 1 gives 0.33, which is a third, and the reason this page prints two decimals rather than four. The fifth and sixth rows are the same pyramid at two scales, and the last row is the Great Pyramid at Giza, whose base of 529000000 square centimetres is 52900 square metres — 230 metres on a side — and whose height of 14660 centimetres is 146.6 metres. That row is in cubic centimetres because that is the page's unit; divided by a million it is 2.585 million cubic metres of stone. The final row is all zeros, which is a pyramid with no base or no height and is an answer rather than an error. Every cell is recomputed from its row when the page is built, in square centimetres and cubic centimetres.

Formula

V = A × h ÷ 3

A
The area of the base, in square centimetres. It is an area rather than a pair of edge lengths because the base can be any polygon: the formula uses the base only through how much ground it covers, so a square base of 36 square centimetres and a hexagonal base of 36 square centimetres give the same volume at the same height. Working the area out is a separate step, and the area calculators for rectangles, triangles, circles and regular polygons are all on this site
h
The height, in centimetres: the perpendicular distance from the base up to the apex, not the distance along a sloping edge. This is the one measurement on a pyramid that cannot be taken with a tape along the surface, and it is the one people get wrong — the length of a corner edge is always longer than the height, because it is the hypotenuse of a triangle whose other sides are the height and half the base
V
The volume, in cubic centimetres. It is a third of what a prism of the same base and height would hold, which is the whole content of the formula: three copies of the pyramid fill the prism exactly, whatever shape the base is. It is printed to two decimals rather than four because the division by three is the one operation in this family that turns whole numbers into repeating decimals — a base of 1 and a height of 1 gives 0.3333 and the useful reading is that it is a third
÷ 3
The constant that makes a pyramid a pyramid. It comes from the fact that three congruent pyramids assemble into a prism of the same base and height, so the pyramid is exactly one third of the prism — not approximately, and not only for square bases. It is the same division as on the cone volume page, since a cone is a pyramid whose base happens to be a circle

Use this page when you have a pyramid and you want to know how much it holds or how much material is in it, and when you can put a number on the base as an area. That last condition is the one that decides whether this is the right page. If the base is a square or a rectangle and you know its two sides, the arithmetic is short enough to do in your head and the area calculators on this site will give you the base in one step; if the base is a hexagon or a pentagon, or an irregular outline you have measured some other way, this page is where that area goes in. The formula is also the fastest sanity check on any pyramid-shaped object: whatever the base, the shape holds a third of what a box of the same footprint and height would hold. That is worth applying before believing any figure — a pyramid-shaped roof over a 100 square metre footprint at 9 metres tall encloses 300 cubic metres of space, where the same roof pushed up into a flat-topped box would enclose 900, and the difference between those two numbers is why pyramid roofs look the way they do. Two smaller uses are worth mentioning. Cutting a pyramid out of solid material is a volume question about waste: a pyramid carved from a block leaves two thirds of the block behind, which is the plainest difference between cutting a shape out of material and casting it. And filling a pyramid — a hopper, a grain bin, a cone-shaped pile of gravel, a sand timer — is this page with the base as the floor area, where the answer is the capacity and the same answer is what a cone page would give for a round one. If your pyramid is not straight-sided, or the apex does not sit over the middle of the base, the page still applies: the volume formula uses the perpendicular height and the base area only, and a leaning pyramid with the same base and the same height holds exactly the same as an upright one, which is a genuinely surprising fact and a useful one.

Worked examples

  1. A base of 36 square centimetres and a height of 4

    1. Base times height: 36 × 4 = 144
    2. Divide by three: 144 ÷ 3 = 48

    The input the page loads with, and the row that shows the relationship the formula is built on: a prism with the same base and height would hold 144 cubic centimetres, and the pyramid holds exactly a third of that. Both numbers are worth keeping, because going from one to the other is a single multiplication or division and the prism figure is usually much easier to sanity-check against the size of the object.

  2. A base of 1 and a height of 1

    1. Base times height: 1 × 1 = 1
    2. Divide by three: 1 ÷ 3 = 0.33333…, which rounds to 0.33

    The unit pyramid, and the reason this page prints two decimals where the cuboid page prints four. One divided by three is 0.3333 to four decimals, which looks like a measurement that came out at three tenths of a thousand; 0.33 reads as what it is, a third of something. That is the whole argument for the narrower width here: the division by three is the only operation in this family that reliably produces a repeating decimal, and more digits of it add no information.

  3. A base of 100 and a height of 9

    1. Base times height: 100 × 9 = 900
    2. Divide by three: 900 ÷ 3 = 300

    The row to check the whole page against, because all three numbers are exact and the prism version is obvious: a box with a base of 100 square centimetres and a height of 9 holds 900 cubic centimetres, and the pyramid holds 300 of them. If a pyramid-shaped roof covered 100 square metres at 9 metres tall it would enclose 300 cubic metres of space, which is the sort of figure that decides how much air a building has to heat.

  4. The Great Pyramid at Giza

    1. Base area in square centimetres: 230 m × 230 m = 52900 m², which is 529000000 cm²
    2. Height in centimetres: 146.6 m = 14660 cm
    3. Base times height: 529000000 × 14660 = 7755139999999.999…, which is 7.75514 × 10¹²
    4. Divide by three: 7755140000000 ÷ 3 = 2585046666666.666…, which rounds to 2585046666666.67

    The largest of the Egyptian pyramids, with its original base of about 230 metres on a side and its original height of about 146.6 metres. The answer is in cubic centimetres because that is the page's unit, and it is not a figure anyone reads directly: divide by a million and it is 2.585 million cubic metres, which is about 2.6 million cubic metres of stone. This is the row where the division by three is worth noticing — the box that would contain the pyramid holds 7.76 million cubic metres, exactly three times the pyramid, so the same footprint and the same height give three times the volume.

Limitations

The base has to be flat and the sides have to run straight from its edges up to a single apex: a pyramid with a stepped profile, like the earlier Egyptian pyramids and the ziggurats, is a stack of boxes and slabs rather than one pyramid, and this formula will be wrong for it by a wide margin. A frustum — a pyramid with its top cut off, which is what most roofs and hoppers actually are — is also not this shape; the volume there needs both the top and bottom areas, and this page has one base field. The height must be the perpendicular distance from the base plane up to the apex, not the distance along a sloping edge or a face; measuring the sloping edge instead is the most common way to get a figure that is too large, and it is too large by a lot on a tall pyramid rather than a little. The base area is an input rather than something the page works out, so a base measured as two edges has to be converted to an area first, and an irregular base has to be measured by whatever method suits it. The page cannot give a slant height, the surface area, or the length of any edge, because none of those can be derived from the base area alone: a long thin base and a compact one with the same area have different slant heights, different surface areas and different edges. The output is in cubic centimetres whatever the base and height dropdowns are set to, so a base entered in square metres comes back as a cubic centimetre answer you have to convert yourself — a million cubic centimetres to the cubic metre, or a millionth of a cubic metre per cubic centimetre. Two decimals is a display width rather than a claim about precision, and it is honest for a measured pyramid but very coarse for a large one, where the last two digits before the decimal point are the only meaningful ones. The page assumes the apex is somewhere over the base and does not care where: an upright pyramid and a leaning one with the same base and the same height have the same volume, which is correct and is worth knowing rather than a limitation. It also has nothing to say about weight, material cost or the waste produced by cutting the shape out of a solid block, all of which can be worked out from the volume but none of which is printed here.

Frequently asked questions

Why is the volume of a pyramid one third of the prism around it?
Because three pyramids of the same base and height fit exactly into the prism, with no gaps and no overlap, whatever polygon the base is. It is not an approximation and it is not specific to square bases — a triangle-based pyramid and a hexagon-based pyramid are both exactly a third of the prism that would contain them. The same one-third appears on the cone volume page, since a cone is a pyramid with a circle for its base.
Do I enter the base's sides or its area?
Its area. The formula uses the base only through how much ground it covers, so the two edges of a rectangular base would be an unnecessarily narrow way to ask. Entering the area also lets the page handle bases it could not otherwise describe: a hexagonal base, a triangular one, or an irregular outline you measured some other way. If your base is a rectangle or a square, the rectangle and square area calculators on this site will give you the area in one step.
What height do I measure?
The perpendicular height: the straight-line distance from the plane of the base up to the apex, measured at right angles to the base. Not the length of a sloping edge, and not the distance up the middle of a face. The sloping edge is always longer than the height — it is the hypotenuse of a triangle whose other sides are the height and half the base's diagonal — so measuring it gives an answer that is too big, and on a tall pyramid much too big.
How much stone is in the Great Pyramid of Giza?
Its original base was about 230 metres square, which is 52900 square metres, and its original height was about 146.6 metres. Base area times height over three gives about 2.6 million cubic metres. The useful comparison is with the box that would contain it, which holds three times as much — 7.76 million — and that gap is the whole point of the division: the same footprint and the same height, a third of the volume.
Can the page give me the slant height or the surface area?
No, and the reason is the input. Slant height is the distance from the apex down to the middle of an edge, and it depends on the shape of the base rather than only on its area: a long thin base of 100 square centimetres and a compact one of the same area have quite different slant heights and quite different surface areas. To compute either, the page would have to know the base's outline, which is exactly what taking an area instead of two edges avoids.
Does a leaning pyramid hold less than an upright one?
No, exactly the same, as long as the base area and the perpendicular height are the same. The volume formula uses only those two numbers and never asks where the apex is above the base, so a pyramid whose peak leans off to one side holds precisely as much as one standing straight over the middle. This is a real result rather than a rounding effect, and it is the same reason a stack of coins holds the same whether or not the coins are perfectly aligned.

References

Related calculators

Cone Volume CalculatorVolume, slant height and surface area of a cone.Cuboid Volume CalculatorVolume and surface area of a rectangular box.Cylinder Volume CalculatorVolume of a cylinder, plus the areas of its ends and its curved side.Sphere Volume CalculatorHow much space a sphere takes, from its radius or diameter.