Quick answer: the volume of a square pyramid is V = (1/3) a² h — base edge squared, times height, divided by three. Measured a slant instead? Vertical height = √(s² − (a/2)²) for the face slant, or √(e² − a²/2) for the corner edge.
The square pyramid formula, step by step
where a = base edge · h = vertical height (base center to apex)
Worked example — a pyramid with a 6-unit base edge and 10-unit height:
- Base area: 6² = 36 square units
- Times height: 36 × 10 = 360
- One-third: 360 ÷ 3 = 120 cubic units
Three measurements you might have instead of h, and how to convert each:
| You measured… | Where it runs | Convert with |
|---|---|---|
| Face slant height (s) | Base midpoint → apex, up a face | h = √(s² − (a/2)²) |
| Lateral edge (e) | Base corner → apex | h = √(e² − a²/2) |
| Face angle (θ) | Tilt of a face from horizontal | h = (a/2) × tan θ |
The benchmark: Giza by the numbers
The Great Pyramid's original dimensions: base edge 756 ft, height 481 ft. Volume = (1/3) × 571,536 × 481 ≈ 91.6 million ft³. Its faces slope at 51.8°, which the angle formula above confirms: (756/2) × tan 51.8° ≈ 480 ft. The formula and the monument agree to within rounding — after 4,500 years.
Where square pyramid volume shows up in real life
Hip roof peaks and finials
A pyramid roof over a 12 × 12 ft gazebo rising 5 ft encloses (1/3) × 144 × 5 = 240 ft³ — the air a ceiling fan in the peak actually has to move.
Hopper bottoms
Square bins drain through inverted square pyramids. A 4-ft-square hopper tapering over 2.5 ft holds (1/3) × 16 × 2.5 ≈ 13.3 ft³ in the taper. If it ends in a square chute instead of a point, subtract the small pyramid that's missing — or use the general pyramid calculator for both parts.
Paperweights and packaging
A pyramid-shaped gift box with an 8-cm base and 9-cm height holds (1/3) × 64 × 9 = 192 cm³ — about a fifth of a liter, which is why pyramid boxes look bigger than they pack.
Common mistakes & pro tips
- Slant height in the h slot. The single most common error on this shape. Slant is always longer than vertical height; convert first with the table above.
- Base perimeter instead of edge. If you measured around all four sides, divide by 4 before entering a.
- Forgetting the ÷3. a²h alone is the enclosing box — three times too much.
- Pro tip — apex off-center? Volume doesn't change. As long as the apex sits at height h above the base plane, (1/3)a²h holds even for leaning pyramids.
Frequently asked questions
What is the formula for the volume of a square pyramid?
V = (1/3)a²h, where a is the base edge and h the vertical height. A pyramid with a 6 cm base edge and 10 cm height holds (1/3) × 36 × 10 = 120 cm³.
How do I convert slant height to vertical height?
For the face slant height s (up the middle of a triangular face): h = √(s² − (a/2)²). For the lateral edge e (along a corner): h = √(e² − a²/2). A pyramid with a 10-unit base and 13-unit slant height has h = √(169 − 25) = 12.
What is the volume of the Great Pyramid?
With its original 756-ft base edge and 481-ft height: V = (1/3) × 756² × 481 ≈ 91.6 million ft³ — about 2.6 million cubic meters of stone.
Why divide by 3?
Every pyramid holds exactly one-third of the prism sharing its base and height. Three identical oblique square pyramids can literally be assembled into one cube — a physical proof you can print and fold.