A right square or rectangular pyramid is fully defined by its base dimensions and its perpendicular height. From those values alone, you can derive the base area, the slant height of each face, the lateral and total surface area, and the volume. This calculator does that math instantly and shows how each value depends on the others.
How the Pyramid Calculator works
The calculator starts from the volume formula V = ⅓ × base area × h, which says a pyramid holds exactly one-third the volume of a prism sharing the same base and height. Base area is simply length × width. Because a rectangular base isn't symmetric, the four triangular side faces come in two matching pairs, each with its own slant height — found with the Pythagorean theorem from the perpendicular height and the apothem (half of the opposite base dimension). Adding up all four triangular faces gives the lateral surface area; adding the base area gives the total surface area.
Inputs and what they mean
Choose Square base to enter a single length (the width mirrors it automatically), or Rectangular base to enter length and width independently. Then enter the height (h) — the straight vertical distance from the center of the base to the apex, not the slant height. All three inputs must be positive numbers in the same unit; the calculator does not convert between units, so pick inches, feet, centimeters, or meters up front and use it consistently.
Limits and edge cases
This calculator assumes a right pyramid with a rectangular (or square) base — the apex sits directly above the center of the base. An oblique pyramid, where the apex is off-center, has different (more complex) slant heights, though the volume formula V = ⅓ × base area × h still applies as long as h is the perpendicular height. For a triangular or other polygonal base, or for a frustum (a pyramid with the tip sliced off), use a dedicated calculator for that shape instead — this tool assumes a four-sided base coming to a single point.