slant height

Stand a paper cone on a table. Two different 'heights' describe it: the straight-up distance from the table to the tip (the vertical height), and the distance measured along the sloping surface from the rim up to the tip (the slant height). They are not the same, and confusing them is one of the most common errors in solid measurement.

For a right cone of radius r and vertical height h, the slant height L is the hypotenuse of a right triangle whose legs are r and h: L = sqrt(r^2 + h^2), straight from the Pythagorean theorem. For a regular pyramid, the slant height is the height of one triangular side face — the distance from the apex straight down the middle of a face to the midpoint of a base edge — and it relates the vertical height to the apothem of the base by the same Pythagorean triangle. The slant height is the longer of the two; only when the apex is directly overhead does the right-triangle relation hold cleanly.

The reason it matters is that slant height, not vertical height, drives the lateral surface area. A cone's curved side unrolls into a sector of radius L, giving lateral area pi · r · L; a pyramid's triangular faces each have height equal to the slant height, giving lateral area (1/2) · (perimeter) · (slant height). Volume, by contrast, uses the vertical height h. So a single problem often needs both: h for volume, L for surface area — keep them straight.