How this instrument works
A cone's slant length is the distance along its curved outer surface, from the apex down to any point on the base's rim — a genuinely different measurement from the perpendicular rise, which runs straight up the cone's own central axis instead. Slicing the cone directly through its apex and center reveals a right triangle: the perpendicular rise as one leg, the base radius as the other, and the slant length as the hypotenuse connecting them, giving slant = √(r² + h²) directly from the Pythagorean theorem.
This distinction matters because the two lengths serve different formulas entirely: a cone's VOLUME uses the perpendicular measurement (⅓πr²h), while a cone's LATERAL SURFACE AREA uses the slant length instead (πr·slant) — mixing the two up is a common, easy-to-make error whenever both quantities show up in the same problem.
The identical relationship — hypotenuse from two perpendicular lengths — reappears throughout this site's other pyramid and cone pages, wherever a slanted face needs relating back to a shape's flat, perpendicular dimensions: it's simply the Pythagorean theorem, applied consistently to whichever right triangle a particular solid's geometry happens to hide.
- Enter the cone's base radius into the Base radius field.
- Enter the cone's perpendicular height (straight up the central axis) into the Perpendicular height field.
- Read Slant height: the sheet applies the Pythagorean theorem directly.
Worked example — radius 3, height 4
A cone has a base radius of 3 and a perpendicular height of 4. Its slant height is √(9+16) = √25 = 5 — a direct 3-4-5 right triangle, with the slant height standing in as the hypotenuse connecting the cone's apex to the rim of its base.
A cone with radius 0 has collapsed to a straight vertical line, and the slant length simply equals the perpendicular rise exactly, 4, since there's no base radius left to angle across at all. A larger cone with radius 6 and a rise of 8 gives a slant length of exactly 10 — the 6-8-10 triangle, the same 3-4-5 triple scaled by 2.
Questions
What is a cone's slant height?
The distance along the cone's curved outer surface, from the apex down to a point on the base's edge — distinct from the perpendicular rise, which instead runs straight up the cone's central axis to the apex directly above the base's center.
How is slant height calculated?
slant = √(r² + h²), where r is the base radius and h is the perpendicular rise. This is the Pythagorean theorem applied to the right triangle formed by slicing the cone through its apex and center.
Why does the distinction between slant length and perpendicular rise matter?
Because different cone formulas need a different one of the two: volume uses the perpendicular measurement, while lateral surface area uses the slant length instead. Substituting the wrong one into either formula gives an incorrect result.
What if the radius is zero?
The cone has collapsed to a straight vertical line, and the slant length equals the perpendicular rise exactly, since there's no base radius left to create any angle between the two.
Does this same relationship apply to pyramids too?
Yes — a pyramid's own slant height (used for its lateral surface area) relates to its perpendicular height and its base's apothem through the identical Pythagorean relationship, just with the base's apothem standing in for a cone's radius.