Slant Height of a Cone

Q: How do you calculate the slant height of a cone?

Use $l = \sqrt{r^2 + h^2}$, where $r$ is the radius and $h$ is the height. Square each value, add them, and take the square root.

The slant height of a cone is the distance from the apex, the pointed tip, down to the edge of the circular base, measured along the outer surface of the cone. It is different from the height, which runs straight down through the middle of the cone. This calculator finds the slant height, radius, or height of a right circular cone from the other two values.

What is slant height?

A right circular cone has a flat circular base and a pointed apex directly above the center of that base. Three measurements describe its shape:

Radius (r) — the distance from the center of the base to its edge.
Height (h) — the straight, perpendicular distance from the base to the apex.
Slant height (l) — the distance from the apex to the edge of the base, measured along the slanted surface.

These three lengths form a right triangle. The height is one leg, the radius is the other leg, and the slant height is the hypotenuse, the longest side, which sits opposite the right angle.

Slant height formula

Because the radius, height, and slant height form a right triangle, the Pythagorean theorem applies:

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

Here $l$ is the slant height, $r$ is the radius, and $h$ is the height.

The formula can be rearranged to find either of the other two lengths, provided the slant height and one more value are known.

To find the radius:

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

To find the height:

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

How to calculate the slant height of a cone

Measure or note the radius and height of the cone, using the same unit for both (for example, both in centimeters).
Square each value, then add the two results together.
Take the square root of that sum. The result is the slant height, in the same unit.

Worked example

A cone has a radius of 3 units and a height of 4 units. To find the slant height:

l = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ units}

The slant height is 5 units.

The same idea works in reverse. A cone has a slant height of 13 units and a height of 12 units. To find the radius:

r = \sqrt{13^2 - 12^2} = \sqrt{169 - 144} = \sqrt{25} = 5 \text{ units}

The radius is 5 units.

Valid values

Radius, height, and slant height must all be positive numbers. A cone with a radius or height of zero has no volume and is not a real three-dimensional shape.

The slant height must always be longer than both the radius and the height. This follows directly from the Pythagorean theorem: the hypotenuse of a right triangle is always longer than either leg. If a proposed slant height is shorter than or equal to the radius or height entered, no valid cone exists with those measurements, and the calculator reports an invalid input rather than showing an incorrect number.

Why slant height matters

Slant height is used to find the lateral surface area of a cone, the curved surface excluding the base, using the formula $\pi r l$ . This matters for tasks such as:

Cutting sheet metal or paper into the correct sector shape to form a cone, as in funnels, paper cups, or lampshades.
Estimating material for conical roofs, spires, or silos.
Solving geometry problems that involve the Pythagorean theorem in three dimensions.

Frequently asked questions

What is the slant height of a cone? It is the distance from the apex of a cone to the edge of its base, measured along the slanted surface, rather than straight down through the center.

How do you calculate the slant height of a cone? Use $l = \sqrt{r^2 + h^2}$ , where $r$ is the radius and $h$ is the height. Square each value, add them, and take the square root.

What is the difference between slant height and height? Height is the straight, perpendicular line from the base to the apex. Slant height runs along the outer surface of the cone from the apex to the base edge, so it is always the longer of the two.

Can the slant height be shorter than the radius or height? No. Because slant height is the hypotenuse of a right triangle formed with the radius and height, it is always longer than either one.

Does this formula work for any cone? It applies to a right circular cone, one with a circular base and an apex centered directly above it. An oblique cone, where the apex sits off to one side, needs a different calculation.

What units should be used? Any unit works, as long as the radius and height are both entered in the same unit. The result comes out in that same unit.

Slant Height of a Cone Calculator