Square Pyramid Volume Calculator
Calculate the volume of a square pyramid from its base edge and height.
About
Square Pyramid Volume Calculator
The volume of a square pyramid is V = a²h/3, where a is the base edge and h the perpendicular height. Enter both above and the calculator shows the working.
What the formula means
A square pyramid has a square base and four triangular faces meeting at an apex. Its volume is the base area, a², multiplied by the height, then divided by three.
The third is not particular to pyramids with square bases. Every pyramid and every cone occupies exactly one third of the prism sharing its base and height, whatever the base looks like. Three identical pyramids can be assembled into the prism they came from, which is the demonstration usually done with plastic models in a classroom.
A worked example
Take a base edge of 2 units and a height of 3.
- Square the base edge: 2² = 4
- Multiply by the height: 4 × 3 = 12
- Divide by three: 4 cubic units
The enclosing box, 2 by 2 by 3, holds 12, and the pyramid holds a third of it. That ratio is the quickest sanity check available.
Height, slant height and lateral edge
A square pyramid has three different lengths that all get called height, and mixing them is the main source of error.
| Measurement | Runs from | Used for |
|---|---|---|
| Height (h) | base centre to apex, vertical | volume |
| Slant height (l) | midpoint of a base edge to apex | face area |
| Lateral edge | a base corner to apex | frame lengths |
They are related by Pythagoras. The slant height is √(h² + (a/2)²), and the lateral edge is √(h² + (a√2/2)²). For a base of 2 and a height of 3, the slant height is √(9 + 1) = 3.162 and the lateral edge is √(9 + 2) = 3.317.
Only the vertical height belongs in the volume formula. Using the slant height would overstate the answer by about 5% in this example, and much more for a squat pyramid.
Surface area
The four triangular faces each have area ½al, so the lateral surface is 2al. Adding the square base gives a total of a² + 2al.
For a base of 2 and a slant height of 3.162, the lateral surface is 2 × 2 × 3.162 = 12.65 and the total with the base is 16.65 square units. Surface area is the figure you need for cladding or covering, and it is where the slant height earns its keep.
The Great Pyramid as a check
The Great Pyramid of Giza has a base of about 230.4 m and an original height near 146.6 m. Its volume works out at 230.4² × 146.6 / 3, which is roughly 2.594 million cubic metres.
At an average block density near 2.6 tonnes per cubic metre that is some 6.7 million tonnes of stone. Estimates of the number of blocks cluster around 2.3 million, which implies an average block volume near 1.1 m³. The volume calculation is what turns a shape into those quantities, and it is how the labour involved gets estimated.
Truncated pyramids
Cut the top off and you have a frustum, the shape of a hopper, a lampshade or a step in a stepped pyramid. Its volume is h(A₁ + A₂ + √(A₁A₂))/3, where A₁ and A₂ are the two square areas.
For squares of 4 m and 2 m sides with a height of 3 m, the areas are 16 and 4, and the volume is 3(16 + 4 + 8)/3 = 28 m³. The formula reduces to the pyramid one when the top area is zero, which is a useful check.
Pyramids around the world
The square pyramid appears independently across cultures, and the reason is structural rather than symbolic. A pile of stone is stable when its sides slope inward, and the square base is the easiest to set out with ropes and right angles.
| Structure | Base | Height | Approximate volume |
|---|---|---|---|
| Great Pyramid, Giza | 230.4 m | 146.6 m | 2,594,000 m³ |
| Pyramid of the Sun, Teotihuacan | 223.5 m | 65.5 m | 1,090,000 m³ |
| Louvre Pyramid, Paris | 35.4 m | 21.6 m | 9,020 m³ |
The Teotihuacan pyramid has a base close to Giza's but under half the height, so it holds well under half the volume. Height drives the figure once the base is fixed, because volume is linear in height and quadratic in the base edge.
Hoppers and storage
A square hopper is a frustum, and its capacity decides how much material a machine can hold between refills. The taper also has to be steep enough that the contents flow out under gravity rather than arching across the opening.
The angle needed depends on the material, and it is generally steeper than the angle of repose. Free-flowing grain moves at around 30 degrees from vertical, while damp powders may need 60 degrees or more, which is why hoppers for difficult materials are so much taller for the same capacity.
Estimating without a ladder
The height of a pyramid you cannot climb can be found from its shadow. When a vertical stick of known height casts a shadow of known length, the ratio applies to everything else at that moment.
A 1 m stick casting a 0.8 m shadow means heights are 1.25 times shadow lengths. A pyramid whose shadow extends 40 m beyond its base edge, plus half the base of 20 m, gives a total shadow of 60 m from the centre and a height near 75 m. Thales is credited with measuring the Great Pyramid this way, and the method needs nothing but a stick and level ground.
Why the third is not obvious
A cone being a third of its cylinder and a pyramid a third of its prism both look like results that ought to have a simple picture behind them, and for the pyramid there is one, though only for a particular case.
A cube can be cut into exactly three identical square pyramids, each with one face of the cube as its base and the cube's centre-opposite corner as its apex. Assemble the three and the cube reappears. That demonstration is exact and needs no calculus, but it only covers pyramids of that specific proportion.
For a pyramid of any other height the general proof integrates the area of horizontal slices. A slice at a fraction t of the way up has edge a(1 − t), so its area is a²(1 − t)². Integrating that square from 0 to 1 gives a third, which is where the factor comes from and why it holds for every base shape.
Volume against material
A solid pyramid and a hollow one need entirely different calculations, and confusing them is expensive. The volume formula gives the space enclosed. A hollow pyramid, such as a roof or a glass structure, needs the surface area instead, which is a² + 2al for a closed one or 2al for the four faces alone.
The Louvre pyramid encloses about 9020 m³ but is built from roughly 1000 m² of glass panels. The enclosed volume tells you about the space inside; the surface area tells you what it cost to clad. Quoting one when the other is wanted is a common source of confusion in project estimates.
Common mistakes
Using slant height for volume. Volume needs the vertical height. Convert with h = √(l² − (a/2)²).
Forgetting the third. Leaving it out gives the enclosing box, three times too large.
Treating a hopper as a full pyramid. Most are frustums with the tip removed.
Mixing units. Base edge and height must share a unit.
Common questions
Frequently asked questions
Base edge squared, multiplied by the perpendicular height, divided by three. A pyramid with a 2 unit base and 3 unit height holds 4 cubic units.
Three identical pyramids assemble into the prism sharing their base and height. The rule holds for any base shape, which is why cones follow it too.
Height runs vertically from the base centre to the apex and is used for volume. Slant height runs from the midpoint of a base edge to the apex and is used for face area. The slant is always longer.
Slant height is the square root of the height squared plus half the base edge squared. For a base of 2 and height of 3, that is the square root of 10, about 3.162.
The base plus four triangles: a squared plus 2 times a times the slant height.
About 2.594 million cubic metres, from a 230.4 m base and a 146.6 m original height. At roughly 2.6 tonnes per cubic metre that is around 6.7 million tonnes of stone.
Use the frustum formula: the height times the sum of both areas plus the square root of their product, all divided by three.
Yes, provided the perpendicular height is used. A tilted pyramid holds the same as an upright one with the same base and height.