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 that often replace 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 |
Volume of a square-based pyramid from any two measurements
Many problems give the base perimeter, the base area or the slant height instead of the edge and height. Choose the pair you have, and the solver converts it and returns the volume with every other measurement.
Square pyramid solver: any two measurements
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Pick the pair you know. The solver returns the base edge, height, slant height, lateral edge, volume and surface areas.
From the base perimeter and height
A square has four equal sides, so the base edge is the perimeter ÷ 4. A pyramid with a 19.8 m base perimeter and a 24.2 m height has a = 4.95 m, a base area of 24.5025 m², and V = 24.5025 × 24.2 ÷ 3 = 197.7 m³ to the nearest tenth.
From the base area and height
Skip the edge entirely: V = base area × height ÷ 3. A base of 8.3 m² and a height of 6.3 m give 8.3 × 6.3 ÷ 3 = 17.4 m³.
Find the height from the volume
Rearrange to h = 3V ÷ a². A square pyramid of 400 cm³ on a 10 cm base stands 1,200 ÷ 100 = 12 cm tall. For the base edge instead, use a = √(3V ÷ h).
Square pyramid surface area
Each triangular face has an area of a × s ÷ 2, so the four faces cover 2as and the total surface area is a² + 2as. A 6 cm base with a 5 cm slant height has 60 cm² of faces and 96 cm² in total.
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.
How do you find the volume of a square-based pyramid from its perimeter?
Divide the base perimeter by 4 to get the edge, square it for the base area, then multiply by the height and divide by 3. A 19.8 m perimeter and a 24.2 m height give a = 4.95 m and V = 4.95² × 24.2 ÷ 3 = 197.7 m³.
What is the surface area of a square pyramid?
Add the square base to the four triangular faces: A = a² + 2as, where s is the slant height up the middle of a face. A pyramid with a 6 cm base and a 5 cm slant height has 36 + 60 = 96 cm² of surface.