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Cone Slant Height Calculator

Compute a cone's slant height instantly

How it is calculated

In a right circular cone the axis is perpendicular to the base, so the radius r, the vertical height h and the slant height l form a right triangle with l as the hypotenuse: l = √(r² + h²).

Enter radius and height in the same unit; the slant height comes out in that unit, rounded to two decimals. Both values must be above zero, otherwise no result appears. The radius is of the base circle (half the diameter), and the height is measured straight up from the centre of the base, not along the side. The formula is valid only for a right cone, not an oblique one whose apex is off-centre.

Worked example

A cone has a base radius of 6 cm and a height of 8 cm. r² = 36 and h² = 64, so r² + h² = 100. The square root of 100 is 10, so the slant height is 10.00 cm. This is the classic 6-8-10 right triangle, which makes it easy to check by hand.

Try this example →

Common mistakes

  • Entering the diameter instead of the radius: the base width is twice the radius, so the slant height comes out too large.
  • Typing the slant height already measured along the side into the height field: height is the vertical distance, and the tool would then compute a longer hypotenuse.
  • Mixing units, such as radius in centimetres and height in metres: both must use the same unit.
  • Using the formula for an oblique cone: it only holds when the apex is directly above the centre of the base.

About this calculator

The slant height of a right circular cone is found with the Pythagorean relationship l = √(r² + h²), where r is the base radius and h is the perpendicular height from the base to the apex — the slant height itself is the straight-line distance along the cone's surface from the edge of the base up to the tip.

It's the figure you actually need for real-world jobs like calculating the lateral surface area of a cone (πrl) or cutting a paper or sheet-metal template that wraps into a cone shape — traffic cones, funnels, lampshades and party hats are all built from a flat sector whose radius equals the slant height, so getting it right determines whether the pattern actually closes into a cone.

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