Calculate the volume, lateral area, total surface area, and slant height of square, rectangular, and triangular pyramids. Includes famous pyramid presets and a reference table.
A pyramid is a polyhedron formed by connecting a polygonal base to a single apex point. Pyramids have fascinated humans for millennia — from the Great Pyramid of Giza, one of the Seven Wonders of the Ancient World, to modern glass pyramids like the Louvre entrance in Paris.
The volume of any pyramid is given by V = ⅓ × Base Area × Height, regardless of the shape of the base. This elegant formula means a pyramid occupies exactly one-third the volume of a prism with the same base and height. For surface area calculations, you need the slant height — the distance from the midpoint of a base edge to the apex, measured along a triangular face.
This calculator supports three common base types: square, rectangular, and equilateral triangular. For a square pyramid with base side a and height h, the slant height is l = √((a/2)² + h²), the lateral area is 2al, and the total surface area adds the base area a². Rectangular and triangular pyramids follow similar principles with adjusted base geometry.
Pyramid geometry appears throughout architecture, packaging design, optics (triangular prisms), and crystal structures. Whether you're designing a rooftop, calculating fill material for a pyramidal mold, or studying ancient monuments, this calculator computes volume, surface area, slant height, and more from your dimensions.
Pyramid Volume & Surface Area problems often require several dependent steps, and a small arithmetic slip can propagate through every derived value. This calculator is tailored to that workflow: you enter base side b, pyramid height, base type, and it returns volume, base area, lateral surface area, total surface area in one consistent pass. It is useful for homework checks, worksheet generation, tutoring walkthroughs, and fast field/design estimates where you need reliable geometry results without rebuilding the full derivation each time.
Volume: V = ⅓ × Base Area × h Square base area: A = a² Rectangular base area: A = a × b Equilateral triangle base area: A = (√3/4) × a² Slant height (square): l = √((a/2)² + h²) Lateral area (square): L = 2al Total surface area: S = L + Base Area
Result: Volume ≈ 2,592,276 m³, Slant height ≈ 186.4 m
The Great Pyramid of Giza has a square base with side 230.4 m and height 146.5 m. Volume = ⅓ × 230.4² × 146.5 ≈ 2,592,276 m³. Slant height = √((230.4/2)² + 146.5²) ≈ 186.4 m.
This pyramid volume & surface area tool links the entered values (base side b, pyramid height, base type, unit) to the target geometry relationships used in class and practice problems. Instead of solving each intermediate step manually, you can validate setup and arithmetic quickly while still tracing which measurements drive the final result.
Formula focus: the calculator formula
Pyramid Volume & Surface Area shows up in school geometry, technical drafting, construction layout checks, and early engineering design estimates. When values are changed repeatedly, the calculator helps you compare scenarios quickly and see how sensitive the shape is to each dimension.
Start with the primary outputs (volume, base area, lateral surface area, total surface area) and then use the remaining cards/tables to confirm consistency with your diagram. Keep units consistent across inputs, and round only at the end if your assignment or project specifies a fixed precision.
V = ⅓ × Base Area × Height. This works for any pyramid shape — square, rectangular, triangular, or any polygonal base.
For a square pyramid with base side a and height h: slant height l = √((a/2)² + h²). It's the distance from the midpoint of a base edge to the apex along a face.
Lateral surface area includes only the triangular side faces. Total surface area adds the bottom base area as well.
The volume formula V = ⅓Bh still applies to oblique pyramids (where the apex is not centered). However, the slant heights of individual faces differ, making surface area more complex.
A pyramid whose base is a regular polygon (all sides equal) and whose apex is directly above the center of the base. All lateral faces are congruent isosceles triangles.
With a volume of about 2.59 million cubic meters and limestone density around 2,600 kg/m³, the Great Pyramid contains roughly 6.7 million tonnes of stone.