The height of a cone calculator will help you find the height of any cone given just two parameters.

Here you will learn:

  • How to find the height of a cone given its volume and radius.
  • How to find the height of a cone without its volume, knowing its radius and slant height.

Is the radius of a cone proportional to its height? Keep reading to learn the answer to that question and read on some cone height examples!

Cone definition

A cone is a three-dimensional shape with a circular base, and a single vertex called the apex. This is the most intuitive cone to picture in your head (such as traffic cones or ice-creams).

The height of a cone calculator works with cones where its apex is located directly above its base center. These are called right circular cones. Cones with apex not above its base center are called oblique cones.

Height of a cone formula

There are two different formulas to find the height of a cone. Given its radius (rr) and slant height (ll):

h=(l2r2)h = \sqrt{(l^{2}-r^{2})}

And given, again, its radius and volume (VV):

h=3×Vπ×r2h = \frac{3\times V}{\pi \times r^{2}}

Let's see when to apply each next.

How to find the height of a cone without knowing its volume?

To find the height of a cone without knowing its volume:

  1. Write down the radius and slant height dimensions.
  2. Input them in the height of a cone formula: h = √(l² - r²) where:
    • l is the slant height;
    • r is the radius; and
    • h is the resulting height.
  3. That's it!

How to find the height of a cone given its volume?

To find the height of a cone given its radius and volume:

  1. Write down the radius and volume.
  2. Input them in the height of a cone formula for volume: h = 3 × V/(π × r²) where:
    • V is the cone's volume;
    • r is the radius; and
    • h is the resulting height.
  3. It's as simple as that!

Examples using the height of a cone calculator

Example 1: Find the height given radius and slant height

Let's say we want to find the height of a cone with radius r=5 cmr = 5\ \text{cm} and slant height l=8 cml = 8\ \text{cm}. Then we use the height of a cone formula without volume:

h=(8 cm)2(5 cm)2h=396.25 cm\begin{align*} h &= \sqrt{(8\ \text{cm})^{2}-(5\ \text{cm})^{2}} \\ h &=\sqrt{39} ≈ 6.25\ \text{cm} \end{align*}

Example 2 Find the height given radius and volume

Now, let's assume the volume of a 20 cm20\ \text{cm} radius cone is V=1 L=1000 cm³V = 1\ \text{L} = 1000\ \text{cm³}.

Looking at the formula from the previous section, we know that the height will be equal to:

h=3×1000 cm³π×(20 cm)2h2.39 cm\begin{align*} h &= \frac{3 \times 1000\ \text{cm³}}{\pi \times(20\ \text{cm})^{2}} \\\\ h & ≈ 2.39\ \text{cm} \end{align*}

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FAQ

Is the radius of a cone proportional to its height?

No. The radius of a cone and the cone's height are independent of each other if there are no fixed variables (for example, the cone's volume). However, the height of a cone and radius are directly proportional to its slant height dimension.

What's the height of a 10 cm radius and 15 cm slant height cone?

5√5 = 11.18. To find the height of a 10 cm radius and 15 cm slant height cone, you need to input those parameters in the height of a cone formula h = √(l² - r²) where:

  • l is the cone's slant height; and
  • r is the radius.
Luciano Mino
Diagram of a right circular cone.
Radius (r)
in
Slant height (l)
in
Volume (V)
cu in
Height (h)
in
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