The slant height is the diagonal distance from the apex of a cone or pyramid down to the edge of the base — measured along the surface of the shape, not straight down through the middle. It always forms the hypotenuse of a right triangle with the vertical height and the base radius (or apothem), and you need it any time a problem asks about lateral surface area.
In both cases, slant height is the hypotenuse — always the longest of the three measurements.
The slant height formula calculates the distance from the apex of a solid figure down to a point on the edge of its base. Unlike the vertical height — which measures the straight distance from top to bottom — the slant height follows the slope of the lateral face.
In a right circular cone, the slant height (l), the vertical height (h), and the radius (r) form a right-angled triangle, so the formula comes straight from the Pythagorean theorem: l = √(r² + h²). When you need the surface area of a cone, this is the first step — the lateral area is π·r·l.
For a regular square pyramid the triangle is formed using the vertical height (h) and half the base side (s/2). So the formula is l = √((s/2)² + h²).
The SAT never gives slant height directly — it makes you find it from height and radius. Recognize the right triangle.
SAT problems often give lateral surface area and ask for a missing dimension. Rearrange the formula.
The SAT's geometry section increasingly includes 3D shapes since going digital. Slant height appears 1–2× per test.
For pyramids, the right triangle uses s/2 (half the base), not the full side length. Always halve it first.
The lateral edge (apex to a base corner) is longer than the slant height. Slant height goes to the midpoint of a base edge.
Volume always uses vertical height (h). Only surface area uses slant height (l).
$l = \sqrt{r^2 + h^2}$, where r is the base radius and h is the vertical height.
Vertical height goes straight down through the center; slant height runs along the sloped face. Slant height is always longer.
Use slant height for lateral surface area; use vertical height for volume.
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