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Cone Slant Height

Also known as slant height of a cone · cone side length

l=r2+h2l = \sqrt{r^2 + h^2}

Worked example: r = 3 m, h = 4 m → slant 5 m — press Try an example to run it live, then adjust anything.

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Cone Slant Height explained

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Slice a right circular cone straight down through its apex and the cut face is an isosceles triangle. Half of it is a right triangle with legs rr and hh and hypotenuse ll, so Pythagoras hands over l=r2+h2l = \sqrt{r^2 + h^2} with no further work. Every cone formula that mentions surface uses ll; every cone formula that mentions volume uses hh, and this is the bridge between them.

In practice you measure whichever is easier and convert. A tape laid up the outside of a silo roof gives you the slant directly. A laser measuring the peak above the eaves gives you the vertical height. Roofers, tent makers and conveyor designers all live on this conversion, and the useful sanity check is that ll must always exceed both rr and hh. If a supplied slant height comes out shorter than the radius, someone has mixed up which measurement is which.

Cone Slant Height formula

l=r2+h2l = \sqrt{r^2 + h^2}
Where
  • ll= Slant height (m)
  • rr= Base radius (m)
  • hh= Vertical height (m)

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