Square Pyramid Volume Calculator — V = ⅓a²h
Enter the side length of the square base and the vertical height, and the calculator applies V = ⅓a²h to find the pyramid's volume — showing every arithmetic step so you can follow the method.
V = ⅓ × base area × height = ⅓a²h
Volume formula
Base area
Multiply by height
Divide by 3
V = 108
- 1
Base area
6² = 36 - 2
Base area × height
36 × 9 = 324 - 3
Volume
324 ÷ 3 = 108Any pyramid holds one-third the volume of a prism with the same base and height.
How does this calculator work?
Volume of a square pyramid = (1/3) × a² × h, where a is the base side and h is the vertical height. The one-third factor comes from the pyramid's tapering — it holds exactly one-third the volume of a square prism with the same base and height. Enter both values to get the volume with step-by-step working.
Formula
How this is calculated
The volume of any pyramid equals one-third of its base area times its vertical height. For a square pyramid with base side a and height h, the base area is a² (a square), so the formula becomes V = (1/3) × a² × h.
The factor of one-third is a mathematical constant shared by ALL pyramids and cones — a pyramid of any cross-section holds exactly one-third the volume of a prism with the same base and height. You can prove this with calculus (integrating cross-sectional areas from base to apex) or with Cavalieri's principle.
This calculator uses the perpendicular height — the straight-line distance from the apex directly down to the base plane. If you know the slant height instead, first recover h from h = √(l² − (a/2)²), where l is the slant height. For the full set of dimensions (surface area, lateral edge, slant height) see the companion Square Pyramid Calculator.
Frequently asked questions
Any pyramid with a given base and height holds exactly 1/3 the volume of a prism with the same base and height. This can be proved by slicing both solids into infinitely thin cross-sectional layers and comparing (Cavalieri's principle), or by integration — the cross-section at height z has area a²(1 − z/h)² and integrating from 0 to h gives a²h/3.
Yes — first convert slant height l to vertical height using h = √(l² − (a/2)²), where a/2 is the distance from the centre of the base to the midpoint of a base edge. Then enter h in the calculator.
The volume is in cubic units matching your input. If you enter the base side and height in centimetres, the volume is in cm³. In metres → m³; in feet → ft³. Make sure both inputs use the same unit.
Also known as
TG we-Calculate Editorial Team. (2026). Square Pyramid Volume Calculator — V = ⅓a²h [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/square-pyramid-volume-calculator
TG we-Calculate Editorial Team. "Square Pyramid Volume Calculator — V = ⅓a²h." TG we-Calculate. 2026. https://we-calculate.com/calculator/square-pyramid-volume-calculator.
TG we-Calculate Editorial Team, "Square Pyramid Volume Calculator — V = ⅓a²h," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/square-pyramid-volume-calculator
@misc{wecalculate_square_pyramid_volume_calculator, title = {Square Pyramid Volume Calculator — V = ⅓a²h}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/square-pyramid-volume-calculator}}, year = {2026}, note = {TG we-Calculate} }
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