# How to find the height of a cone?

#### Understand the Problem

The question is asking for a method or formula to determine the height of a cone, which is a geometric figure. It may involve using the formula for the volume or surface area of the cone, along with given measurements such as the radius or slant height.

Use $h = \sqrt{l^2 - r^2}$ or $h = \frac{3V}{\pi r^2}$.

To find the height (h) of a cone, you can use the Pythagorean theorem if the radius (r) and the slant height (l) are known: $$h = \sqrt{l^2 - r^2}$$. Alternatively, if the volume (V) and the radius (r) are known, you can use the volume formula rearranged to: $$h = \frac{3V}{\pi r^2}$$.

#### Steps to Solve

1. Identify the given measurements of the cone

Typically, you need at least two of the following measurements: the radius (r), the height (h), the slant height (l), or the volume (V). Let's assume we have the radius and the slant height given.

1. Use the Pythagorean theorem to find the height

A cone forms a right triangle when the height (h), radius (r), and slant height (l) are considered. According to the Pythagorean theorem:

$$l^2 = r^2 + h^2$$

1. Solve for the height (h)

Rearrange the equation to solve for the height:

$$h^2 = l^2 - r^2$$

Then, take the square root of both sides to find h:

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

This formula will give you the height (h) of the cone when the radius (r) and the slant height (l) are known.

1. Alternative method if the volume is given

If the volume (V) of the cone and the radius (r) are given, you can use the volume formula of a cone:

$$V = \frac{1}{3} \pi r^2 h$$

Rearrange to solve for the height (h):

$$h = \frac{3V}{\pi r^2}$$

This formula will give you the height (h) of the cone when the volume (V) and the radius (r) are known.

To find the height (h) of a cone, you can use the Pythagorean theorem if the radius (r) and the slant height (l) are known: $$h = \sqrt{l^2 - r^2}$$. Alternatively, if the volume (V) and the radius (r) are known, you can use the volume formula rearranged to: $$h = \frac{3V}{\pi r^2}$$.