Cone Slant Height Calculator
Find the slant height of a right circular cone from its radius and vertical height using l = √(r² + h²), plus the lateral surface area.
How it works
l = √(r² + h²) · Lateral = π·r·l · Total = π·r·l + π·r²
The slant height is the straight-line distance from the tip of a right circular cone down to the edge of its base. The radius, the vertical height and the slant height form a right triangle, so the slant height is the hypotenuse: l = √(r² + h²) by the Pythagorean theorem. It is the length you need for the cone’s lateral (curved) surface area, π·r·l.
Worked example
For a base radius of 3 and a vertical height of 4, l = √(3² + 4²) = √(9 + 16) = √25 = 5 units, giving a lateral surface area of π·3·5 = 15π ≈ 47.1239 square units.
Sources: Cone — Wikipedia
Frequently asked questions
Is the slant height the same as the vertical height?
No. The vertical height is the perpendicular distance from the apex straight down to the centre of the base; the slant height runs along the surface to the base edge and is always longer (unless the radius is zero).
Why is the Pythagorean theorem used?
The radius, vertical height and slant height form a right triangle, with the slant height as the hypotenuse. So l² = r² + h², giving l = √(r² + h²).
What is slant height used for?
It is needed for the cone’s lateral surface area, π·r·l, and for the total surface area π·r·l + π·r². The volume, by contrast, uses the vertical height, not the slant height.
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This calculator is for educational purposes. Double-check important results before acting on them.