The surface area of a square pyramid is SA = s squared plus 2sl, the base area plus its four triangular faces, where s is the base side and l is the slant height. Learn where the formula comes from, how to find the slant height with the Pythagorean theorem, the volume formula, and a worked example.
Surface area of a pyramid
The surface area of a pyramid is the total area of all its outer faces: the base plus every triangular side. For a square pyramid with base side s and slant height l (the distance from the middle of a base edge up to the apex), the formula is SA = s² + 2sl — the base area (s²) plus the area of the four triangular faces (2sl).
Unfolding a square pyramid into its net shows the base plus four triangular faces, each with slant height l.
Where the formula comes from
Each triangular face has area ½×base×height = ½×s×l. A square pyramid has 4 identical triangular faces, so their combined area is 4×(½sl) = 2sl. Adding the square base (s²) gives the full formula, SA = s² + 2sl.
Volume of a pyramid
The volume of any pyramid is V = ⅓×base area×height, where height is the straight-up distance from the base to the apex (not the slant height). A pyramid always holds exactly one-third the volume of a prism with the same base and height — a relationship worth remembering when you compare it to the volume of a cylinder.
Finding the slant height
If you're only given the pyramid's straight-up height (h) and the base's half-side, you can find the slant height with the Pythagorean theorem: l² = h² + (s/2)². This comes up often, since problems more commonly give you the vertical height than the slant height directly.
Example
For a square pyramid with base side s = 6 and slant height l = 5: SA = 6² + 2(6)(5) = 36 + 60 = 96 square units. Its volume, with height h = 4, is V = ⅓(6²)(4) = ⅓(144) = 48 cubic units.