What is a square pyramid's surface area?
A square pyramid has a square base and four identical triangular faces that meet at a single apex. Its total surface area is the sum of the square base area and the combined area of those four triangles. This calculator computes the total surface area, base area, lateral (sides) area, and slant height from just two measurements: the base edge length and the vertical height.
How to use it
Enter the base edge length b (the side of the square base) and the pyramid height h (the perpendicular distance from base to apex). Both values must use the same units. The calculator returns the total surface area in square units, along with a breakdown of base area, lateral area, and slant height.
The formula explained
The base area is simply \(b^2\). Each triangular face has a base of \(b\) and a height equal to the slant height \(l\), where \(l = \sqrt{\left(\frac{b}{2}\right)^2 + h^2}\). The four faces together give a lateral area of \(2\cdot b\cdot l\). Adding them yields:
$$A = b^2 + 2\,b\,\sqrt{\left(\frac{b}{2}\right)^2 + h^2}$$
The slant height comes from the right triangle formed by half the base edge, the pyramid height, and the slant itself.
Worked example
Suppose \(b = 4\) and \(h = 6\). First find the slant height:
$$l = \sqrt{\left(\frac{4}{2}\right)^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} \approx 6.3246$$
Base area \(= 4^2 = 16\). Lateral area \(= 2 \times 4 \times 6.3246 \approx 50.596\). Total surface area \(\approx 16 + 50.596 = \) 66.596 square units.
FAQ
What is slant height vs. pyramid height? The pyramid (vertical) height goes straight up from the base center to the apex. The slant height runs along a triangular face from the midpoint of a base edge to the apex, and it is always longer than the vertical height.
Does this work for any pyramid? This calculator is specific to a square pyramid (square base, four equal triangular faces). Rectangular or triangular pyramids use different formulas.
What units does it use? Any consistent linear unit — the result is in those units squared (e.g. cm² if you entered cm).
