Slant Height — Definition, Formula & Examples

Slant Height

The diagonal distance from the apex of a right circular cone or a right regular pyramid to the base.

Two diagrams: a right circular cone and a right regular pyramid (square base), each labeled with height and slant height.

See also

Volume, surface area

Key Formula

l = \sqrt{h^2 + r^2}

Where:

$l$ = Slant height of the cone (or pyramid)
$h$ = Vertical height from the base to the apex
$r$ = Radius of the base (for a cone) or the distance from the center of the base to the midpoint of a base edge (apothem, for a regular pyramid)

Worked Example

Problem: A right circular cone has a vertical height of 12 cm and a base radius of 5 cm. Find the slant height.

Step 1: Identify the known values. The vertical height is 12 cm and the base radius is 5 cm.

h = 12, \quad r = 5

Step 2: Notice that the vertical height, the radius, and the slant height form a right triangle. The slant height is the hypotenuse.

l^2 = h^2 + r^2

Step 3: Substitute the values into the Pythagorean theorem.

l^2 = 12^2 + 5^2 = 144 + 25 = 169

Step 4: Take the square root of both sides to find the slant height.

l = \sqrt{169} = 13 \text{ cm}

Answer: The slant height of the cone is 13 cm.

Another Example

This example applies slant height to a pyramid instead of a cone. For a pyramid, you use the apothem of the base (distance from center to midpoint of a side) in place of the radius.

Problem: A right regular square pyramid has a vertical height of 9 m and each side of the square base measures 24 m. Find the slant height of the pyramid.

Step 1: Find the apothem of the square base—this is the distance from the center of the square to the midpoint of one side. For a square with side length s, the apothem is half the side length.

a = \frac{s}{2} = \frac{24}{2} = 12 \text{ m}

Step 2: The vertical height, the base apothem, and the slant height form a right triangle. The slant height runs from the apex down the center of a triangular face to the midpoint of the base edge.

l^2 = h^2 + a^2

Step 3: Substitute the known values.

l^2 = 9^2 + 12^2 = 81 + 144 = 225

Step 4: Take the square root to solve for the slant height.

l = \sqrt{225} = 15 \text{ m}

Answer: The slant height of the pyramid is 15 m.

Frequently Asked Questions

What is the difference between slant height and height?

The height (also called the vertical or perpendicular height) is the straight-line distance from the apex directly down to the center of the base. The slant height is the diagonal distance along the outer surface from the apex to the edge of the base. The slant height is always longer than the vertical height because it is the hypotenuse of the right triangle they share.

When do you use slant height?

You use slant height when calculating the lateral (side) surface area of a cone or pyramid. For a cone, the lateral surface area formula is

A = \pi r l

, where

l

is the slant height. For a regular pyramid, the lateral surface area is

A = \frac{1}{2} P l

, where

P

is the perimeter of the base. Volume formulas, by contrast, use the vertical height.

How do you find slant height if you know the surface area?

Rearrange the lateral surface area formula. For a cone with lateral area

A_L

and radius

r

, solve

l = \frac{A_L}{\pi r}

. For a regular pyramid with lateral area

A_L

and base perimeter

P

, solve

l = \frac{2 A_L}{P}

. This is useful in reverse problems where surface area is given and you need to find slant height.

Slant Height vs. Vertical Height

	Slant Height	Vertical Height
Definition	Diagonal distance from apex to the base edge along the lateral face	Perpendicular distance from the apex straight down to the center of the base
Symbol commonly used	l (or sometimes s)	h
Formula (cone)	l = √(h² + r²)	h = √(l² − r²)
Used for	Lateral surface area calculations	Volume calculations
Length comparison	Always longer (hypotenuse)	Always shorter (leg of the triangle)

Why It Matters

Slant height appears frequently in geometry courses whenever you calculate the lateral surface area of cones and pyramids. Without it, you cannot correctly find how much material covers the sides of these shapes—an essential skill in problems involving paint, fabric, roofing, and packaging. It also reinforces the Pythagorean theorem in a three-dimensional context, bridging the gap between 2D and 3D reasoning.

Common Mistakes

Mistake: Using the vertical height instead of the slant height in lateral surface area formulas.

Correction: Lateral surface area formulas (

\pi r l

for cones,

\frac{1}{2}Pl

for pyramids) specifically require the slant height

l

, not the vertical height

h

. Always check which measurement the formula calls for.

Mistake: For pyramids, using the full base side length instead of the apothem when calculating slant height.

Correction: The right triangle is formed by the vertical height and the apothem (center of base to midpoint of a side), not by the full side length or the distance to a vertex. For a square base with side

s

, the apothem is

s/2

Related Terms

Apex — The top point from which slant height is measured
Right Circular Cone — Primary 3D shape where slant height applies
Right Regular Pyramid — Other common shape using slant height
Base — The bottom face to which slant height extends
Surface Area — Lateral surface area requires slant height
Volume — Uses vertical height, not slant height
Pythagorean Theorem — The formula used to derive slant height