How this instrument works
A square pyramid has a square base and four triangular faces meeting at a single apex directly above the base's center. Its volume follows the same rule every pyramid and cone shares: one third of the base area times how tall it stands — the pyramid always holds exactly a third of the volume of the square prism (box) built on the same base and the same vertical rise.
The surface area needs one more step: the base's own square area, plus four identical triangular faces. Each triangular face's own slant measurement (distinct from the pyramid's straight vertical rise) comes from a right triangle formed by that vertical rise and half the base edge — the Pythagorean theorem applied to a cross-section straight through the pyramid's apex.
This is a distinct solid from the rectangular boxes and cylinders already covered elsewhere on this site — a pyramid's triangular faces and single apex give it a genuinely different volume relationship (a third of the equivalent prism, rather than the prism's full volume) worth seeing spelled out on its own.
- Enter the pyramid's base edge length into the Base side length field.
- Enter its vertical height into the Height field.
- Read Volume: a third of the base area times the height.
- Read Total surface area: the base plus all four triangular faces, each found via the slant height.
Worked example — base 6, height 4
A square pyramid with a 6-unit base edge and a height of 4 has a volume of 6²×4⁄3 = 48 cubic units — a third of the 144 cubic units the equivalent square prism (a box 6 by 6 by 4) would hold. Its slant height is √(4²+3²)=5, a 3-4-5 right triangle formed from the height and half the base edge (3), giving a total surface area of the base (36) plus four triangles (2×6×5=60), for 96 square units total.
A larger pyramid with a 10-unit base and height 12 gives a volume of 100×12⁄3=400, and a slant height of √(12²+5²)=13 (a 5-12-13 triangle), for a total surface area of 100+2×10×13=360.
Questions
What is the formula for the volume of a square pyramid?
V = b²h⁄3, where b is the base edge length and h is the vertical height — exactly one third of the volume of a square prism (box) built on the same base and the same height.
What is the slant height, and how is it different from the height?
The height is the straight vertical distance from the base to the apex; the slant height runs along the surface of a triangular face instead, found via the Pythagorean theorem from the height and half the base edge — it's always somewhat longer than the vertical height itself.
Why is a pyramid's volume a third of the equivalent prism's?
It's a general geometric result true of every pyramid and cone: shrinking from the base up to a single point at the apex always accounts for exactly a third of the volume the same base and height would enclose if extended straight up into a prism or cylinder instead.
Does the surface area formula include the base?
Yes — the total reported here is the base's own square area (b²) plus the four triangular side faces combined (2b times the slant height), giving every exposed surface of a solid pyramid resting on a table.
What if the height is very small compared to the base?
The pyramid flattens toward a thin, wide shape — the slant height approaches half the base edge itself, and the volume shrinks toward zero even though the base area stays the same, since volume depends directly on the height.