What the Square Pyramid Volume Calculator Does
This calculator finds the volume of a square pyramid — a solid with a square base and four triangular faces that meet at a single apex. You only need to enter two measurements: the Base Edge (a), which is the length of one side of the square base, and the Height (h), the perpendicular distance from the centre of the base straight up to the apex. The tool instantly returns the volume, and behind the scenes it also computes the base area, slant height and total surface area for context.
The Formula Explained
The volume is calculated using the standard geometry formula:
$$V = \frac{1}{3} \cdot \text{Base Edge (a)}^{2} \cdot \text{Height (h)}$$
Here a² is the area of the square base, and multiplying by the height and one-third gives the volume. A pyramid always holds exactly one-third the volume of a prism (box) with the same base and height — that is why the 1/3 factor appears.
The calculator also derives some useful extras from your two inputs:
- Base area = \(a^2\)
- Slant height = \(\sqrt{h^2 + (a/2)^2}\) — the distance from the apex down the middle of a triangular face
- Surface area = \(a^2 + 2 \times a \times \sqrt{(a/2)^2 + h^2}\) — the base plus the four triangular sides
Worked Example
Suppose your pyramid has a base edge of 6 units and a height of 9 units.
- $$\text{Volume} = \frac{1}{3} \times 6^2 \times 9 = \frac{1}{3} \times 36 \times 9 = 108 \text{ cubic units}$$
- Base area = \(6^2 = 36\) square units
- Slant height = \(\sqrt{9^2 + 3^2} = \sqrt{90} \approx 9.49\) units
- Surface area = \(36 + 2 \times 6 \times 9.49 \approx 149.9\) square units
Just match the units you enter (cm, m, inches) and the volume comes out in the cubed version of that unit.
Frequently Asked Questions
Do I use height or slant height? Use the perpendicular height (h) — the straight vertical line from the base centre to the apex. Slant height runs along the face and is longer; using it would overstate the volume.
What units should I use? Any unit works as long as both inputs share it. If you enter centimetres, the volume is in cubic centimetres (cm³).
Does this work for non-square pyramids? No. This tool assumes a perfectly square base where all four base edges equal a. For rectangular bases you would need a different formula, \(V = \frac{1}{3} \times \text{length} \times \text{width} \times h\).
