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How to use
This calculator handles a right square pyramid: a square base with the apex directly over its center. Base side and vertical height determine the face slant height, corner edge, areas and volume.
- Enter base side a and vertical height h in the same unit.
- Face slant height l is computed from a right triangle with legs h and a/2.
- The apex-to-corner lateral edge is calculated separately so it is not confused with l.
- Review volume, base area, lateral area and total surface area.
How it is calculated
Example
Example: a=6 and h=4 gives slant height 5, volume 48, lateral area 60 and total area 96; the apex-to-corner edge is ≈5.831.
Important notes
The lateral-area formula assumes a right square pyramid. Oblique or rectangular-base pyramids need different face geometry, although V=Bh/3 still uses perpendicular height.
Worked examples and interpreting results
A right square pyramid with base side 6 cm and perpendicular height 4 cm has volume 48 cm³. Its face slant height is 5 cm and is used for surface area; volume uses perpendicular height.
| Case | Calculation | Result |
|---|---|---|
| Volume | 6² × 4 ÷ 3 | 48 cm³ |
| Face slant height | √(4² + (6/2)²) | 5 cm |
| Total surface area | 6² + 2 × 6 × 5 | 96 cm² |
How to check the result
Face slant height differs from the edge joining the apex to a base corner. These equations place the apex above the base center; an oblique pyramid needs other information for its face areas.
Frequently asked questions
Slant height versus lateral edge?
Slant height runs from the apex to the midpoint of a base edge and is used for face area; the lateral edge runs to a base corner and is longer.
Why divide volume by 3?
Any pyramid has one third the volume of a prism with the same base area and perpendicular height.
Does it work for a rectangular-base pyramid?
No. This calculator is deliberately scoped to a right square base.