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Surface area of a square-based pyramid

2026-09-03Spatial GeometryMs Ho

Add the base to four triangular faces — or use slant height for a quicker lateral total.

Key formulas

total SA = a² + 2aℓ
ℓ = slant height of a face

Pieces

Base a². Each lateral face is a triangle with base a and slant height ℓ (the height of the triangular face, not the pyramid height). Four faces give 2aℓ.

Example

a=6, ℓ=5 → SA = 36 + 60 = 96.

Finding ℓ

If perpendicular height h is given, ℓ = √(h² + (a/2)²) for a square base.

Worked example

Worked example

Problem. A square-based pyramid has base side a = 6 cm and face slant height ℓ = 5 cm. Find the total surface area.

Solution. Base = 36. Lateral = 2aℓ = 2×6×5 = 60. Total SA = 36 + 60 = 96 cm².

Lateral only

Sometimes exams ask for lateral surface — omit a².

Review quiz

Square pyramid a=10, face slant height ℓ=8. Total SA is:

Practice questions

  1. a=10, ℓ=8. Find total SA.
  2. a=8, h=3. Find ℓ then lateral SA.

Study notes

Slant height vs lateral edge?

Slant height meets the midpoint of a base edge; a lateral edge goes to a base vertex.

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