Mathematics July 22, 2026 · 10 min read

Slant Height: Understanding Cone and Pyramid Geometry

Calculate slant height using l = √(r² + h²). Complete guide to cone and pyramid slant height with worked examples.

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Slant height — the distance from the apex of a cone or pyramid to the edge of its base, measured along the surface — is a crucial measurement for calculating lateral surface area, material requirements, and structural properties.

The Formula

l = √(r² + h²)

This is simply the Pythagorean theorem applied to the right triangle formed by the slant height, perpendicular height, and base radius.

Key Distinction

The perpendicular height (h) is the straight-line distance from apex to base center. The slant height (l) is always longer than the perpendicular height because it follows the surface.

Worked Example

Find the slant height of a cone with radius 5 cm and height 12 cm:

  1. Identify dimensions: r = 5 cm, h = 12 cm
  2. Apply Pythagorean theorem: l = √(5² + 12²)
  3. Calculate: l = √(25 + 144) = √169 = 13 cm
  4. Result: The slant height is 13 cm (classic 5-12-13 right triangle)

Relationship to Surface Area

For a cone: Lateral SA = πrl. For a pyramid: Lateral SA = ½ × perimeter × l. The slant height is essential for these calculations.

Applications

  • Architecture: Roof pitch calculations
  • Manufacturing: Material estimation for cone/pyramid shapes
  • Engineering: Structural analysis of tapered elements