Pyramid Volume Calculator
A pyramid on a square base holds exactly a third of the box that would enclose it: (1/3) b² h, the base side squared, times the vertical height, divided by three. Type the base side and the height and the answer appears beside a drawing labelled with your own numbers, the formula on the stage above it and the working, your figures put in, as the first row below: a 3 m base under a 1.2 m apex reads (1/3) × 3² × 1.2 = 3.6 and gives 3.6 m³, which the rows underneath repeat as 3,600 litres and about 951 US gallons. Height here is the vertical distance from the centre of the base straight up to the apex, never the slanted face, and the dashed line down the middle of the drawing is the one it labels. This page opens the calculator on the Pyramid whatever shape you used last, takes metres, centimetres, feet or inches, and lists litres, US gallons and UK gallons beside cubic feet, cubic yards and cubic inches.
Pyramid · (1/3) b² h3,600 L · 951.019 US gal · 791.889 UK gal
A square base of side b and the height straight up to the apex. A third of the box around it.
The formula is on the stage, and the working is the first row. Every answer here is one textbook formula with your numbers put in, printed so you can check it: a cylinder is π r² h, a sphere is (4/3) π r³, a pool is length times width times the average of its two depths. The horizontal tank uses the exact circular-segment formula for a partial fill, so half way up is exactly half and a quarter of the way up is less than a quarter, because the bottom of a circle is narrow. Conversions use exact definitions: a litre is 1,000 cm³, a US gallon is 231 cubic inches and a UK gallon is 4.54609 litres.
Common questions
- What is the formula for the volume of a pyramid?
- (1/3) b² h for a pyramid standing on a square base: the area of that base, b², times the height, divided by three. The stage prints it as Pyramid · (1/3) b² h, and the first row underneath, The working, repeats it with your numbers in place, so a base of 3 m and a height of 1.2 m read (1/3) × 3² × 1.2 = 3.6 and the answer is 3.6 m³, or 3,600 litres. The b in the formula is one side of the square base, not the area of it and not the distance around it, so a base measuring 3 m along one edge is typed as 3.
- Do I type the vertical height or the slant height?
- The vertical height, straight up from the centre of the base to the apex. The tool does not turn a slant into a height for you, so do that step first with Pythagoras. For a slant height l measured up the middle of a triangular face, h is the square root of l² − (b/2)²: a 10 cm base with a 13 cm slant gives h = 12, because 13² − 5² = 144, and typing 10 and 12 in centimetres reads (1/3) × 10² × 12 = 400 cm³, which the litres row gives as 0.4 L. For an edge e measured from a base corner to the apex, h is the square root of e² − b²/2: an 8 cm base with a 9 cm edge gives h = 7, and the working row reads (1/3) × 8² × 7 = 149.333333333. The right triangle calculator in the related list solves that missing side from the two you measured.
- My pyramid has a rectangular base. Can I still use this page?
- There is no rectangular pyramid on the shape row, but one square root gets the exact answer out of the one that is there: type the square root of length times width as the Base. (1/3) b² h with b as the square root of l × w is (1/3) × l × w × h, the rectangular formula itself rather than an approximation of it. A base 4 m by 9 m under a 3 m apex: the square root of 36 is 6, so type 6 and 3 and the working reads (1/3) × 6² × 3 = 36, giving 36 m³ and 36,000 litres. The drawing keeps showing a square base while you do this, because it draws what the formula takes.
- What about a triangular base, or a tetrahedron?
- The same trick, one step earlier. Every pyramid is a third of its base area times its height, and this page reaches the base area through b², so type the square root of the base area into Base. The area calculator in the related list works a triangle out from a base and a height or from its three sides. A triangular floor of 3 m² under a 3 m apex: the square root of 3 is 1.73205080757, and with a height of 3 the working reads (1/3) × 1.73205080757² × 3 = 3, so 3 m³. A regular tetrahedron of edge a has a base area of a² × √3 / 4 and a height of a × 0.816497, so a 10 cm one is typed as a base of 6.58037 and a height of 8.16497, giving 117.851 cm³, which is what a³ / (6√2) comes to.
- Why is the volume a third of the box the pyramid sits in?
- Because three pyramids of that base and that height stack up to fill the box exactly, and the (1/3) in the formula is that fact. The shape row lets you check it in two clicks: Box with a length and a width of 4 m and a height of 3 m gives 4 × 4 × 3 = 48 m³, and Pyramid with a 4 m base and the same 3 m height gives (1/3) × 4² × 3 = 16 m³, one third of it. Cone does the same to the cylinder around it, (1/3) π r² h against π r² h, which is why a cone and a pyramid of equal base area and height hold the same amount.
- How do I get litres, gallons or cubic yards out of it?
- They are already listed under the answer. Pick metres, centimetres, feet or inches on the Unit row, type the base and the height in that unit (the field labels say which, for instance Base (ft)), and the volume comes back in the cube of it with litres, US gallons and UK gallons underneath every time, plus three cubic units picked for the scale you are working at. A heap of soil shaped to a 12 ft square base and 9 ft high is (1/3) × 12² × 9 = 432 ft³, and the Cubic yards row reads exactly 16, because a cubic yard is 27 cubic feet. Switching the unit re-reads the numbers you already typed rather than converting them, so retype the sides after you change it.
- Why is there no Radius or Diameter switch on this page?
- Because a pyramid has no round part. That switch belongs to the cylinder, the sphere, the hemisphere, the cone and the tank, where a measurement across is easier to take than one from the centre, and it appears only when one of those is selected. The pyramid asks for two straight measurements instead, Base and Height, and both go into the formula as typed. If a number is missing or is not a measurement the answer waits and a line under the fields says which field to fix, for example Base must be greater than 0.
- How big is the Great Pyramid of Giza?
- Type the figures usually quoted for it, a base of 230.3 m and an original height of 146.6 m, and the working row reads (1/3) × 230.3² × 146.6 = 2591794.66467, about 2.59 million cubic metres of stone. Those two measurements are yours to supply and not something this page knows: it does the arithmetic on whatever you type, exactly, and published surveys of the monument differ by a metre here and there. It is solid rather than hollow, so read the result as bulk; the litres and gallons rows are the same volume said another way rather than a claim about capacity.
Volumes are computed from the formulas shown, exact for the dimensions you type; a horizontal tank's partial fill is the exact circular-segment result. A US gallon is 231 cubic inches and a UK gallon 4.54609 litres.