What Is a Square Pyramid?
A square pyramid is a three-dimensional solid with a square base and four triangular faces that meet at a single apex above the center of the base. It is one of the most familiar pyramid shapes — the Great Pyramids of Giza are square pyramids. This calculator computes the volume, total surface area, slant height, base area, and lateral surface area from just two measurements: the base edge length and the perpendicular height.
How to Use This Calculator
Enter the base edge length (a) — the side of the square base — and the height (h), measured straight up from the center of the base to the apex. The calculator instantly returns all key properties. Make sure both values use the same unit (cm, m, in, ft); results will be in those units, squared for areas and cubed for volume.
The Formulas Explained
The volume is one-third of the base area times the height: $$V = \frac{1}{3} \cdot a^2 \cdot h$$ For surface area we first need the slant height — the distance from the base edge midpoint to the apex along a triangular face — given by the Pythagorean theorem: $$l = \sqrt{\left(\frac{a}{2}\right)^2 + h^2}$$ Each triangular face has area \(\frac{1}{2} \cdot a \cdot l\), and there are four of them, giving lateral area \(2 \cdot a \cdot l\). Adding the square base \(a^2\) yields the total surface area $$SA = a^2 + 2a \cdot l$$
Worked Example
Suppose \(a = 6\) and \(h = 4\). Volume $$= \frac{1}{3}(36)(4) = 48 \text{ cubic units}$$ The slant height $$= \sqrt{3^2 + 4^2} = \sqrt{25} = 5$$ Lateral area \(= 2 \cdot 6 \cdot 5 = 60\). Base area \(= 36\). Total surface area \(= 36 + 60 = 96\) square units.
FAQ
What is the difference between height and slant height? The height (h) is the vertical distance from base to apex; the slant height (l) runs along the surface of a triangular face. They are related by \(l = \sqrt{\left(\frac{a}{2}\right)^2 + h^2}\).
Does this work for any pyramid? No — these formulas assume a right square pyramid (square base, apex directly above the center). Rectangular or oblique pyramids need different formulas.
What units does it use? Any consistent unit. If you enter meters, volume is in cubic meters and areas in square meters.
