Slant Height of Cone Calculator

Calculate the slant height of a cone: l = √(r² + h²). Solve for l, r, or h from any two known values.

Choose What to Solve For

Result

Slant Height (l)
--
units
Radius (r)--
Height (h)--
Slant Height (l)--
Lateral Surface Area--
Total Surface Area--
Volume--
Half-Angle (α)--

Step-by-Step Solution

Cone Slant Height Formula

The slant height of a cone is the distance from any point on the circular base to the apex (tip) of the cone, measured along the surface. It forms the hypotenuse of a right triangle where the two legs are the radius (r) and the perpendicular height (h) of the cone. The formula is derived directly from the Pythagorean theorem.

All Three Formulas

Find Slant Height

Given radius r and height h, compute the slant height l.

l = sqrt(r^2 + h^2)

Find Radius

Given slant height l and height h, compute the radius r.

r = sqrt(l^2 - h^2)

Find Height

Given slant height l and radius r, compute the height h.

h = sqrt(l^2 - r^2)

Lateral Surface Area

The curved surface area of the cone (excluding the base).

LSA = pi * r * l

Total Surface Area

Lateral surface area plus the circular base area.

TSA = pi * r * (r + l)

Volume

One-third of the base area times the height.

V = (1/3) pi r^2 h

Real-World Applications

Cone slant height calculations are used in engineering, architecture, and manufacturing. Ice cream cones, traffic cones, volcanic shapes, rocket nose cones, and funnel designs all require accurate slant height calculations for material estimation, surface area calculations, and structural design.

Important Notes

  • The slant height is always greater than both the radius and the height.
  • When solving for radius or height, the slant height must be larger than the other known value.
  • The half-angle at the apex equals arctan(r/h).
  • Unrolling a cone's lateral surface produces a sector of a circle with radius equal to the slant height.