Slant Height of a Cone Calculator

Enter a cone's base radius and height to get its slant height (l = √(r² + h²)), lateral surface area, total surface area, and base angle.

Quick Facts

Slant height formula
l = √(r² + h²)
The Pythagorean theorem applied to radius and height.
Lateral surface area
A = πrl
Area of the curved side only, not the circular base.
Total surface area
A = πr(r + l)
Lateral area plus the base area πr².

Your Results

Calculated
Slant Height
-
l = √(r² + h²)
Lateral Surface Area
-
A = π × r × l
Total Surface Area
-
A = π × r × (r + l)
Base Angle
-
Angle between slant side and base

Ready

Enter a base radius and height, then press Calculate.

Formula and Method for the Slant Height of a Cone

A right circular cone has a circular base of radius r and an apex directly above the center of that base at perpendicular height h. The slant height, l, is the straight-line distance from the apex down to a point on the edge of the base, measured along the cone's outer surface. Because r, h, and l form a right triangle — r along the base, h straight up the axis, and l as the hypotenuse — the slant height follows directly from the Pythagorean theorem: l = √(r² + h²). This calculator also derives the lateral surface area, total surface area, and the base angle from the same two inputs.

How the calculation works

Enter the base radius and the perpendicular height in the same unit, then choose that unit. The calculator squares both values, adds them, and takes the square root to get the slant height: l = √(r² + h²). It then multiplies π × r × l to get the lateral (side) surface area, adds the base area π × r² to get the total surface area (A = πr(r + l)), and computes the base angle — the angle between the slant side and the base plane — as atan(h ÷ r), converted to degrees.

Common mistakes

  • Confusing height with slant height: the height (h) runs straight up the cone's axis; the slant height (l) runs along the outer surface and is always the longer of the two for any cone with r > 0.
  • Using diameter instead of radius: if you only know the diameter, divide it by 2 before entering the radius — using the full diameter roughly doubles the slant height.
  • Mixing units: enter the radius and height in the same unit (both feet, both centimeters, etc.); the calculator does not convert between them automatically.

Real-world applications

  • Sheet-metal and fabric pattern making uses slant height to lay out the flat "sector" pattern that rolls into a cone (funnels, lampshades, traffic cones, tent canopies).
  • Roofing uses slant height for conical or "witch's hat" turret roofs to calculate shingle or panel area along the sloped surface.
  • Packaging design uses lateral and total surface area (which depend on slant height) to estimate material for cone-shaped containers and cups.
  • Construction and engineering use the base angle to check the pitch of conical structures such as silos, hoppers, and spires.

Frequently Asked Questions

What is the formula for the slant height of a cone?
The slant height is l = √(r² + h²), where r is the base radius and h is the perpendicular height. This comes directly from the Pythagorean theorem applied to the right triangle formed by the radius, the height, and the slant line along the cone's side.
How is slant height different from the height of a cone?
The height (h) is the perpendicular distance from the apex straight down to the center of the base. The slant height (l) is the distance from the apex to a point on the base's edge, measured along the outer surface. Since the slant line is the hypotenuse of the r-h right triangle, the slant height is always longer than the height for any cone with r > 0.
How do you find the slant height if you know the diameter instead of the radius?
Divide the diameter by 2 to get the radius, then apply l = √(r² + h²). For example, a cone with a 10 ft diameter (r = 5 ft) and a 12 ft height has a slant height of √(5² + 12²) = √169 = 13 ft.
How is slant height used to find the lateral surface area?
The lateral (side) surface area of a cone is A = πrl, where r is the base radius and l is the slant height. Adding the base area (πr²) gives the total surface area: A = πr(r + l). Slant height is required for both because the cone's side is a curved surface, not a flat vertical wall.

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