Our slant height of a cone calculator will answer one of the most pressing questions you may ever have: how long is the side of an ice cream cone?

Here you will learn everything you need about cones and their oblique sides, from the base to the apex:

  • What is a cone?
  • Which are the elements of a cone?
  • How to find the slanted height of a cone?
  • Practical examples!

What is a cone?

A cone is a solid figure originated by a complete rotation of a right triangle around one of its catheti.

The resulting solid has a circular base and a continuous slant side that connects the edge of the base to the topmost point of the cone, the apex.

When we talk of cones, we usually mean a half-cone: the noun cone refers to a figure obtained by rotating for 180° a tilted line, creating two open "conical" shapes. However, in this tool, we will talk only about the former kind.

The elements of a cone

A cone is identified by:

  • A circular base with radius rr; and
  • A height hh perpendicular to the base.

We can identify another element, though:

  • A slant side called slant height of the cone, ll
A cone
A cone with all its elements marked.

In the figure, you can easily identify all of the elements mentioned above.

How to find the slant height of a cone

Here we will teach you how to calculate the slant height of a cone. It is pretty easy, you'll see!

The slant height of a cone is nothing but the hypotenuse of the right triangle that generates the cone. We can apply the Pythagorean theorem to calculate the slant height of any cone, knowing the radius and the height.

The formula for the slant height of a cone is:

l=r2+h2l = \sqrt{r^2+h^2}

How to calculate the slant height of cones in other ways

What if you don't know radius and height, but only one of them, plus the angle at the base of the cone? We need to apply some basic trigonometry!

Calling the angle at the base of the cone α\alpha, and the angle at the apex β\beta, two formulas tell you how to find the slant height of a cone:

l=hsinαl=rsinβ\begin{align*} &l = \frac{h}{\sin{\alpha}}\\ \\ &l = \frac{r}{\sin{\beta}} \end{align*}

How to use our slant height of a cone calculator

Using our tool will help you apply the formula for the slant height of a cone whenever you need it!

To use our tool, simply insert the values you have at hand, and find out the value of the slanted height.

What about some examples now? Let's talk ice cream: imagine (or go get!) an ice cream cone with radius 2.5 cm2.5\ \text{cm} and height 15 cm15\ \text{cm}. Insert these values in the appropriate fields of the slant height of a cone calculator. It will apply the formula:

l=r2+h2=2.52+152 cm=15.2 cm\begin{align*} l &= \sqrt{r^2+h^2}\\ &=\sqrt{2.5^2+15^2}\ \text{cm}\\ &= 15.2\ \text{cm} \end{align*}

Pretty similar to the height? The ice cream cone is a pretty slender one!

What about our other favorite cones, the traffic cones? Take a 28 in28\ \text{in} height cone (the best type). Measure its diameter at the base. It can be, let's say, 10.5 in10.5\ \text{in}. Change the units in our tool, and find out the slant height:

l=r2+h2=10.52+282 in=28.49 in\begin{align*} l &= \sqrt{r^2+h^2}\\ &=\sqrt{10.5^2+28^2}\ \text{in} \\ &= 28.49\ \text{in} \end{align*}

An all-around shape

We didn't stop at the slant height of a cone: check out our other tools dedicated to that pointy shape!

FAQ

What is the slant height of a cone?

The slant height of a cone is the measure of the segment connecting the apex of a cone to the outer rim of its base. It corresponds to the length of the hypotenuse of the right triangle that generates the cone itself.

How do I calculate the slant height of a cone?

If you know the height and the radius of a cone, you must apply the Pythagorean theorem to find the length of the slant height of a cone.

  1. Measure the height and radius of your cone;
  2. Apply the formula `l = sqrt(r² + h²), where:
    • l is the slant height of the cone;
    • r is the radius of the base; and
    • h is the height of the cone.
  3. You are all set!

What is the slant height of a cone with radius 10 cm and height 20 cm?

The slant height of such a cone is 22.36 cm. Apply the slant height of a cone formula to find the length of a cone with base radius r = 10 cm and height h = 20 cm:
l = sqrt(r² + h²) = sqrt(10² + 20²) = sqrt(500) = 22.36 cm

What is the slant height of Mount Fuji?

The slant height of Mount Fuji, assuming it to be a geometric cone, is 22.82 km. The volcano has an average base radius r = 22.5 km and we can approximate its height to h = 3.8 km. To climb up Mount Fuji on a straight line, you have to walk:
l = sqrt(r² + h²) = sqrt(22.5² +3.8²) = 22.82 km
Surprisingly, this is pretty close to the length of the actual trail!

Davide Borchia
Diagram of a right circular cone.
Radius (r)
in
Height (h)
in
Slant height (l)
in
Base angle (α)
deg
Apex angle (ß)
deg
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