Pyramid Volume Formula
V = ⅓ × B × h — one third of the base area times the vertical height. A square base of side 6 with a height of 4 gives ⅓ × 36 × 4 = 48 cubic units. It works for every base shape, so only the way you find B changes. Use the vertical height, never the slant height.
One formula, every base
Most people learn a separate pyramid formula for each base shape and end up memorising five of them. There is only one:
V = ⅓ × (area of the base) × (vertical height)
| Base shape | Base area B | Volume |
|---|---|---|
| Square, side a | a² | ⅓a²h |
| Rectangle, l × w | lw | ⅓lwh |
| Triangle, base b, height t | ½bt | ⅙bth |
| Regular hexagon, side a | 2.598a² | 0.866a²h |
| Regular octagon, side a | 4.828a² | 1.609a²h |
| Circle, radius r (a cone) | πr² | ⅓πr²h |
That last row is worth noticing: a cone is just a pyramid with a circular base. It is not a special case needing its own rule — it is the same rule with a different B.
Where the ⅓ comes from
The ⅓ is not a correction factor or an approximation. It is exact, and it holds for every base shape.
A pyramid occupies exactly one third of the prism that shares its base and height. Euclid proved this in the Elements, and you can demonstrate it physically: three identically-shaped pyramids pack together to fill a cube with no gaps and no overlap. Sets of three plastic pyramids sold for exactly this demonstration are a staple of maths classrooms.
| Base | Height | Prism volume | Pyramid volume | Ratio |
|---|---|---|---|---|
| 36 (6 × 6 square) | 4 | 144 | 48 | ⅓ |
| 50 | 12 | 600 | 200 | ⅓ |
| 93.53 (hexagon, a = 6) | 8 | 748.25 | 249.42 | ⅓ |
| 28.27 (circle, r = 3) | 10 | 282.74 | 94.25 | ⅓ |
Why intuition says one half. Looking at a pyramid inside its prism, the empty space around it looks like about the same amount as the pyramid — suggesting a half. It is not. The pyramid's cross-section shrinks as the square of the distance from the apex, so it loses volume far faster going up than a linear taper would. That squaring is exactly what turns ½ into ⅓.
Vertical height, not slant height
This is the single most common error, and it is easy to make because a real pyramid gives you the slant height more readily than the vertical one.
| Measurement | Runs from apex to | Use it for |
|---|---|---|
| Vertical height h | The centre of the base, straight down | Volume |
| Slant height l | The midpoint of a base edge | Surface area |
| Lateral edge e | A corner of the base | Cutting the edge itself |
They are linked by Pythagoras. For a square pyramid of base a: l² = h² + (a/2)², so if you measured the slant height you can recover the vertical one with h = √(l² − (a/2)²).
For a base of 6 and a slant height of 5, the vertical height is √(25 − 9) = 4. Feed 5 into the volume formula instead of 4 and you overstate the answer by 25%.
Worked examples
A square-based pyramid, base 6, height 4:
- Base area = 6² = 36
- Volume = ⅓ × 36 × 4 = 48 cubic units
A rectangular pyramid, base 8 × 6, height 5:
- Base area = 48
- Volume = ⅓ × 48 × 5 = 80 cubic units
The Great Pyramid of Giza — original base about 230.4 m square, height about 146.6 m:
| Step | Working | Result |
|---|---|---|
| Base area | 230.4² | 53,084.16 m² |
| × height | × 146.6 | 7,782,138 m³ |
| × ⅓ | ÷ 3 | 2,594,046 m³ |
| For scale | ÷ 2500 m³ | ≈ 1,038 Olympic pools |
Roughly 2.59 million cubic metres of stone. The internal chambers and passages are negligible against that — they account for far less than a tenth of one percent.
Leaning pyramids and Cavalieri's principle
If the apex is not directly above the centre of the base — an oblique pyramid — the volume formula is completely unchanged.
This follows from Cavalieri's principle: two solids with the same height and matching cross-sectional area at every level have the same volume. Sliding the apex sideways shifts each horizontal slice but does not change its area, so the total is identical.
A useful mental image is a stack of cards. Push the stack over into a lean and it still contains exactly the same number of cards. Surface area does change, though — the faces become longer and unequal — so an oblique pyramid needs more material to cover despite holding the same amount.
Cutting the top off
A pyramid with its tip removed is a frustum, and averaging the two base areas does not work. The correct formula is:
V = ⅓ × h × (A₁ + A₂ + √(A₁A₂))
The √(A₁A₂) term — the geometric mean of the two ends — is what the naive average misses. Because the cross-section shrinks as the square of the distance from the apex, area does not change linearly along the height, so a linear average always overstates the volume.
The truncated pyramid calculator handles this, and the same structure applies to a truncated cone.
Common mistakes
| Mistake | Effect | Fix |
|---|---|---|
| Using slant height as vertical height | Overstates — 25% in the worked case | h = √(l² − (a/2)²) |
| Forgetting the ⅓ | 3× too big — that is the prism | Pyramids always take a third |
| Using ½ instead of ⅓ | 50% too big | The cross-section shrinks as the square |
| Averaging bases on a frustum | Overstates | Include the √(A₁A₂) term |
| Using the base perimeter as the area | Wrong dimension entirely | B is an area, in square units |
| Assuming a lean changes the volume | Unnecessary correction | Cavalieri — volume is unchanged |
Frequently asked questions
What is the pyramid volume formula?
V = ⅓Bh — one third of the base area times the vertical height. It works for any base shape, so the only thing that changes between a square pyramid and a hexagonal one is how you calculate B. A square base of side 6 with height 4 gives ⅓ × 36 × 4 = 48 cubic units.
Why is there a one third in the pyramid formula?
Because a pyramid occupies exactly one third of the prism sharing its base and height. Euclid proved it, and you can see it physically — three identically-shaped pyramids pack together to fill a cube with no gaps. The ⅓ is an exact geometric fact, not an approximation or fudge factor.
Which height do I use in the pyramid formula?
The vertical height, straight up from the centre of the base to the apex. Not the slant height, which runs down a sloping face and is always longer. Using slant height in the volume formula overstates the answer — for a base of 6 with slant 5, using 5 instead of the true height 4 overstates by 25%. Slant height is for surface area.
Does the formula work for any base shape?
Yes — square, rectangular, triangular, pentagonal, hexagonal, any polygon, and even a circle, which makes it a cone. Only the base-area calculation changes. That universality is why it is worth memorising the general form rather than a dozen specific ones.
What is the volume of the Great Pyramid of Giza?
Using its original dimensions — about 230.4 m square at the base and 146.6 m tall — the volume is ⅓ × 230.4² × 146.6 ≈ 2.59 million cubic metres, or roughly 1,038 Olympic swimming pools. The internal chambers and passages are a negligible fraction of that.
How is a pyramid different from a prism?
A prism has two identical parallel bases joined by flat sides, so its cross-section stays constant and volume is base × length. A pyramid has one base and tapers to a point, so volume is ⅓ × base × height. Same base and height means the pyramid holds exactly a third as much.
What if the apex is not above the centre of the base?
The volume formula is unchanged. An oblique pyramid has exactly the same volume as an upright one with the same base and perpendicular height — this is Cavalieri's principle: solids with matching cross-sectional areas at every level have equal volumes. Think of leaning a stack of cards. Surface area does change, however.
How do I find the volume of a truncated pyramid?
A frustum needs a different formula: V = ⅓h(A₁ + A₂ + √(A₁A₂)). Averaging the two base areas and multiplying by height is a common shortcut and it is wrong — the cross-section shrinks as the square of distance from the apex, so area does not change linearly along the height.