Finding the volume of a cone typically involves knowing the height of the cone. However, in some cases, you might have the slant height and the base radius instead. In this article, we'll walk you through how to determine the volume of a cone using the slant height and base radius by first finding the height.
Formula to Calculate the Volume of a Cone
The volume (\( V \)) of a cone can be calculated using the formula:
\[ V = \pi \cdot r^2 \cdot \frac{h}{3} \]
Where:
- \( V \) is the volume of the cone.
- \( r \) is the radius of the base of the cone.
- \( h \) is the height of the cone.
Finding the Height Using the Slant Height
When the slant height (\( l \)) and the radius (\( r \)) are given, we can find the height (\( h \)) of the cone using the Pythagorean theorem. The relationship between the slant height, height, and radius of the cone is given by:
\[ l^2 = r^2 + h^2 \]
Solving for \( h \):
\[ h = \sqrt{l^2 - r^2} \]
Example Calculation
Let's consider an example to illustrate the process.
Given:
- \( r = 4 \) units (radius of the base)
- \( l = 9 \) units (slant height of the cone)
We aim to find the volume of the cone.
Step 1: Identify the Given Values
Given:
- \( r = 4 \) units
- \( l = 9 \) units
Step 2: Calculate the Height (\( h \)) Using the Slant Height
Use the Pythagorean theorem to find \( h \):
\[ h = \sqrt{l^2 - r^2} \]
Substitute the given values:
\[ h = \sqrt{9^2 - 4^2} \]
\[ h = \sqrt{81 - 16} \]
\[ h = \sqrt{65} \]
Step 3: Substitute the Height into the Volume Formula
Now that we have the height, use it in the volume formula:
\[ V = \pi \cdot r^2 \cdot \frac{h}{3} \]
Substitute \( r = 4 \) units and \( h = \sqrt{65} \) units:
\[ V = \pi \cdot 4^2 \cdot \frac{\sqrt{65}}{3} \]
Step 4: Calculate the Values
First, calculate \( r^2 \):
\[ 4^2 = 16 \]
Next, substitute and simplify:
\[ V = \pi \cdot 16 \cdot \frac{\sqrt{65}}{3} \]
\[ V = \frac{16 \cdot \pi \cdot \sqrt{65}}{3} \]
Step 5: Calculate the Final Value
Using \( \pi \approx 3.14159 \):
\[ V \approx \frac{16 \cdot 3.14159 \cdot \sqrt{65}}{3} \]
Calculate \( \sqrt{65} \):
\[ \sqrt{65} \approx 8.06226 \]
Now calculate the volume:
\[ V \approx \frac{16 \cdot 3.14159 \cdot 8.06226}{3} \]
\[ V \approx \frac{404.840}{3} \]
\[ V \approx 134.947 \]
Final Value
The volume of a cone with a base radius of 4 units and a slant height of 9 units is approximately \( 134.947 \) cubic units.
