Look a little closer.

A square base, a centred apex. These connections help explain why the figure behaves as it does, beyond simply entering numbers into a formula.

01 / A useful relationship

Its volume is one-third that of a prism with the same base and vertical height.

The vertical height, face slant height and lateral edge are different lengths. A face slant height runs from the apex to the midpoint of a base edge. Total surface area is the base plus the four triangular faces.

02 / A different perspective

It has five faces, eight edges and five vertices.

A right square pyramid has a square base and an apex directly above the base centre. Its four lateral faces are congruent isosceles triangles. These assumptions are essential to its slant-height formulas.

03 / A connection to explore

A lateral edge reaches a corner, making it longer than the face slant height.

Use the vertical-height or face-slant tool according to the measurement available. Target volume and total surface area can also determine height when the base edge is fixed.

Try the idea with numbers

Open the base edge and vertical height calculator, note its outputs, then multiply each length input by two. Volume should increase by eight and surface area by four, when every relevant length is doubled. Angle inputs and dimensionless ratios remain unchanged. This scaling experiment checks the structure of the formula rather than the precision of any real measurement.

Further reading: Right square pyramid at Wolfram MathWorld.