Pyramid Volume Calculator
Compute the volume and surface area of a square pyramid from its base edge and height.
Volume
16
Volume = ⅓ × base area × height. A base of 4 with height 3 gives a volume of 16.
Total surface area
44.8444
Base area (a²)
16
Slant height
3.6056
Lateral surface
28.8444
The slant height runs from the apex to the middle of a base edge: √(h² + (a/2)²). Surface area adds the square base to the four triangular faces.
How it works
A square pyramid has a square base and four triangular faces meeting at a point above it. Its volume follows the same one-third rule as any pyramid or cone: volume = ⅓ × base area × height. Since the base is a square, that's ⅓ × a² × h.
Enter the base edge a and the vertical height h. With a base edge of 4 and a height of 3, the base area is 16 and the volume works out to exactly 16 cubic units. The height here is the perpendicular distance from the apex straight down to the center of the base, not the slanted edge.
To get the surface area the tool needs the slant height — the distance from the apex to the middle of a base edge — which it finds from √(h² + (a/2)²). It then adds the square base (a²) to the four triangular faces (together 2 × a × slant) for the total surface area.
Frequently asked questions
What's the difference between height and slant height?
Height is the straight vertical drop from the apex to the center of the base. Slant height runs from the apex down the face to the midpoint of a base edge, so it's always longer. Volume uses the vertical height; surface area uses the slant height.
Why is there a one-third in the volume formula?
A pyramid fills exactly one-third of the prism that shares its base and height. That's why the volume is ⅓ × base area × height rather than base area × height. The same factor shows up in the cone formula for the same reason.
Does this work for non-square pyramids?
This calculator assumes a right square pyramid, where the base is a square and the apex sits directly over its center. For a rectangular or triangular base you'd swap in that base's area and recompute the slant heights of the different faces.