Square Pyramid Volume Calculator
Calculate the volume of a square-based pyramid using base side length and perpendicular height. Formula: V = (1/3) x a^2 x h. Instant and accurate results.
How to use this tool
- Enter base side length and height in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your volume and the full breakdown beneath it.
Formula
V = (1/3) x a^2 x h
How it works
Use V = ⅓a²h. Square the base side, multiply by height, then divide by 3.
Worked example
Pyramid base=6, height=9
- a = 6, h = 9
- a^2 = 36
- V = (1/3) x 36 x 9 = (1/3) x 324 = 108
Volume = 108 cubic units
Common mistakes to avoid
- Using the slant height of a triangular face instead of the perpendicular height h from apex to base — this inflates the volume.
- Forgetting the 1/3 factor and computing a^2 * h instead of (1/3) * a^2 * h, tripling the correct answer.
- Squaring only one dimension when the base is square — the base area is a * a (side squared), not just a.
Key terms
- What is the perpendicular height?
- The vertical distance from the base to the apex, measured at a right angle to the base.
- How is a square pyramid related to a cube?
- Three square pyramids with equal base and height can exactly fill a cube with the same side length.
Frequently asked questions
- What is the perpendicular height of a pyramid?
- It is the straight-line distance from the apex directly down to the centre of the square base, measured at 90 degrees to the base. It is not the slant edge or the slant height of a triangular face.
- Does the formula work for rectangular pyramid bases?
- No. For a rectangular base, replace a^2 with length * width: V = (1/3) * l * w * h. The square pyramid formula is a special case where both base dimensions are equal.
- Why is there a 1/3 factor in the pyramid volume formula?
- A pyramid has exactly one-third the volume of a prism with the same base area and height. This is proven geometrically by decomposing any prism into three equal pyramids.