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Square Pyramid Calculator

Find the volume of a square pyramid.

Result

48.00 units³
96.00 units²

How it is calculated

The tool treats the shape as a right pyramid, whose apex is directly above the centre of a square base. With base edge a and vertical height h: V = a² × h / 3. The slant height of each triangular face is s = √((a/2)² + h²), and the total surface area is the base plus four triangles: A = a² + 2 × a × s.

Both inputs must be greater than zero and in the same length unit; the results are labelled generically as units³ and units², so read them in your own unit (cm³ and cm², for example). Height means the perpendicular height from the base to the apex, not the slanted edge or the slant height of a face.

Worked example

Base edge 6 and height 4. Volume = 6² × 4 / 3 = 36 × 4 / 3 = 48.00 units³. Slant height = √(3² + 4²) = √25 = 5. Surface area = 6² + 2 × 6 × 5 = 36 + 60 = 96.00 units².

Try this example →

Common mistakes

  • Entering the slant height of a face as the height. The tool wants the vertical height; for a slant of 5 over a base of 6, the height is 4.
  • Forgetting the base in the surface. Total surface includes the square base; the four sides alone (lateral area) are 60 here.
  • Mixing units, such as base in centimetres and height in metres, which makes both results wrong.
  • Using the formula for an oblique pyramid. Volume stays the same, but the surface area formula here does not apply.

About this calculator

This calculator finds the volume and total surface area of a right square pyramid from its base edge length a and height h. Volume is V = ⅓ · a² · h — one-third of the base area times height, the same one-third factor that applies to any pyramid or cone. Total surface area adds the square base (a²) to the four triangular lateral faces, each computed from the slant height, l = √(h² + (a/2)²), as ½·a·l per face.

This geometry underlies real structures from ancient monuments like the Great Pyramid of Giza to modern architectural pyramids (the Louvre Pyramid, hotel atriums), and the volume formula is used to estimate material quantities for pyramid-shaped roofs, tents, and packaging.

Geometry students solving solid-shape problems, architects and engineers estimating material for pyramidal structures, and anyone curious about historical pyramid volumes use this calculator by entering the base edge and height.

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