A square pyramid has one square base and four identical isosceles triangles rising to a single apex.
About this calculator
A square pyramid has one square base and four identical isosceles triangles rising to a single apex. Volume equals one-third of the box that shares the same base and height — exactly the same ⅓ rule that applies to cones and all pyramids with flat bases.
Slant height runs from the midpoint of a base edge up the triangular face to the apex. Lateral surface area is four triangles each with area ½ × b × l, which simplifies to 2bl. Add the base b² for total outer area.
Real pyramids, tent peaks, and certain roof caps follow this model when the footprint is square. Compare with [[cone]] for circular bases, [[volume]] for mixed solids, and [[surface-area]] when you only need the wrap area. For ancient-scale examples, plug base 230 m and height 146 m to approximate the Great Pyramid order of magnitude.
Common uses
- Geometry homework — volume from base 6 and height 8
- Roof cap or tent peak material estimates
- Compare pyramid vs rectangular prism volume at equal base
- Surface area for four triangular panels plus square floor
- Scale models and 3D print fill-volume checks
Frequently asked questions
What is the volume when base = 6 and height = 8?
V = ⅓ × 36 × 8 = 96 cubic units. Slant height l = √(3² + 8²) = √73 ≈ 8.54 units.
Why is there a 1/3 in the volume formula?
A pyramid fills exactly one-third of the prism that shares its base and height. Integrating triangular cross-sections from base to apex produces the same factor as for cones.
Is slant height the same as vertical height?
No. Vertical height h is perpendicular to the base. Slant height l runs along the tilted triangular face and is always longer unless the apex sits directly over the base center with zero tilt.
Does this work for rectangular (non-square) bases?
This calculator assumes a square base. For a rectangle b × c, volume = ⅓ × b × c × h and each triangular face may differ.