Slant Height Calculator

Calculate the slant height of cones and pyramids using the Pythagorean theorem.

Select Shape & Enter Dimensions

Result

Slant Height (l)
--
units
Slant Height (l)--
Height (h)--
Base Dimension--
Lateral Surface Area--
Total Surface Area--
Volume--

Step-by-Step Solution

Understanding Slant Height

Slant height is the distance measured along the lateral face of a three-dimensional solid (like a cone or pyramid) from the base to the apex. It differs from the vertical height, which is the perpendicular distance from the base to the apex. The slant height is always longer than the vertical height.

Slant Height Formulas

Cone

The slant height of a cone uses the radius and perpendicular height.

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

Square Pyramid

The slant height runs from the midpoint of a base edge to the apex. Uses half the base edge.

l = sqrt((a/2)^2 + h^2)

Triangular Pyramid

For a regular triangular pyramid, the apothem of the base is a/(2sqrt(3)).

l = sqrt((a/(2sqrt(3)))^2 + h^2)

Pythagorean Theorem

All slant height formulas are based on the Pythagorean theorem: c = sqrt(a^2 + b^2).

l^2 = r^2 + h^2

Practical Applications

Slant height is essential for calculating the lateral surface area of cones and pyramids. In construction, it determines the actual length of material needed for roofing on pyramid-shaped structures. In manufacturing, it helps calculate the amount of material for conical containers, funnels, and party hats.

Tips

  • The slant height is always greater than the vertical height.
  • For a cone, the lateral surface area is pi x r x l.
  • The slant height forms the hypotenuse of a right triangle with the height and base radius.
  • Make sure to use the radius (half the diameter) for cone calculations.