How this calculator works
The slant height is the distance from the rim of the base to the apex, measured along the sloping surface. It forms a right triangle with the vertical height and a horizontal run to the point where the slant meets the base. For a cone that run is the radius. For a square pyramid the slant height runs to the middle of a base edge, so the run is half the base side. The lateral area then uses the slant height: πrl for a cone and 2al for a square pyramid.
Formula
Cone: l = √(r² + h²), lateral area = πrl. Square pyramid: l = √(h² + (a/2)²), lateral area = 2al.
Worked example
A cone with a radius of 3 m and a height of 4 m has a slant height of √25 = 5 m and a lateral area of 15π ≈ 47.124 m². A square pyramid with a base side of 6 m and a height of 4 m also has a slant height of 5 m, with a lateral area of 2 × 6 × 5 = 60 m².
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Frequently asked questions
Is slant height the same as the height?
No. The vertical height is measured straight up from the base to the apex. The slant height follows the sloping surface, so it is always longer for any shape with a non-zero base.
Which base measurement does the pyramid use?
It uses one side of a square base. The slant height is measured to the midpoint of an edge, so the horizontal run is half that side. A rectangular base would have two different slant heights.
When would I need a slant height?
Slant height is needed for the curved or lateral surface area of a cone or pyramid, which is the amount of material for a tent, a roof or a conical cover. It does not appear in the volume formula.