Understand the right square pyramid

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.

Properties that make the calculations work

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.

Where the model is useful

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.

Measure the right thing

Do not substitute a lateral edge for a face slant height. The apex must lie over the base centre for these calculators.

Sketch the figure and mark every known measurement before choosing a calculator. A length measured along an edge is not automatically a perpendicular height. If a formula needs a radius, check that the value has not been measured across a full diameter. Write the unit beside each known quantity before entering it.

Scale and units

Multiplying every length by k multiplies any area by k² and any volume by k³. This is a useful check: doubling all dimensions cannot merely double an area. Inputs expressed in centimetres produce cm² for area and cm³ for volume; the unit selector relabels rather than converts entered numbers.

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Wolfram MathWorld: Right square pyramid provides reference reading. Our summaries and examples are written for this portal and are not a specialist-reviewed textbook. Try a worked example or suggest a correction.