Calculate the square of the radius

Oct 13, 2025 by windailys.com 38 views

What is the Exact Volume of a Cone with a Height of 9 and a Radius of 12?

Understanding the Formula for the Volume of a Cone

The volume of a cone is a fundamental concept in mathematics, and it is essential to understand the formula that calculates this volume. The formula for the volume of a cone is given by:

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

where $V$ is the volume of the cone, $r$ is the radius of the base of the cone, and $h$ is the height of the cone.

Calculating the Volume of a Cone with a Height of 9 and a Radius of 12

To calculate the volume of a cone with a height of 9 and a radius of 12, we can plug these values into the formula:

V = \frac{1}{3} \pi (12)^2 (9)

First, we need to calculate the square of the radius:

(12)^2 = 144

Next, we multiply the result by the height:

144 \times 9 = 1296

Now, we multiply the result by $\frac{1}{3}$ and $\pi$ :

\frac{1}{3} \times 1296 \times \pi = 432 \pi

Conclusion

Therefore, the exact volume of a cone with a height of 9 and a radius of 12 is $432 \pi$ .

Comparison of Options

Let's compare our result with the given options:

A. $392 \pi$ B. $432 \pi$ C. $200 \pi$ D. $60 \pi$

Our result matches option B, which is $432 \pi$ .

Real-World Applications of the Volume of a Cone

The volume of a cone has numerous real-world applications, including:

Architecture: The volume of a cone is used to calculate the volume of a building's dome or a cone-shaped roof.
Engineering: The volume of a cone is used to calculate the volume of a cone-shaped tank or a cone-shaped pipe.
Physics: The volume of a cone is used to calculate the volume of a cone-shaped container or a cone-shaped object.

Tips and Tricks for Calculating the Volume of a Cone

Here are some tips and tricks for calculating the volume of a cone:

Use the formula: The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$ .
Plug in the values: Plug in the values of the radius and height into the formula.
Calculate the square of the radius: Calculate the square of the radius before multiplying it by the height.
Multiply the results: Multiply the results of the square of the radius and the height by $\frac{1}{3}$ and $\pi$ .

Common Mistakes to Avoid

Here are some common mistakes to avoid when calculating the volume of a cone:

Forgetting to square the radius: Forgetting to square the radius before multiplying it by the height.
Forgetting to multiply by $\frac{1}{3}$ and $\pi$ : Forgetting to multiply the result by $\frac{1}{3}$ and $\pi$ .
Using the wrong formula: Using the wrong formula for the volume of a cone.

Conclusion

In conclusion, the exact volume of a cone with a height of 9 and a radius of 12 is $432 \pi$ . The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$ , and it is essential to understand this formula to calculate the volume of a cone.
What is the Exact Volume of a Cone with a Height of 9 and a Radius of 12?

Understanding the Formula for the Volume of a Cone

The volume of a cone is a fundamental concept in mathematics, and it is essential to understand the formula that calculates this volume. The formula for the volume of a cone is given by:

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

where $V$ is the volume of the cone, $r$ is the radius of the base of the cone, and $h$ is the height of the cone.

Calculating the Volume of a Cone with a Height of 9 and a Radius of 12

To calculate the volume of a cone with a height of 9 and a radius of 12, we can plug these values into the formula:

V = \frac{1}{3} \pi (12)^2 (9)

First, we need to calculate the square of the radius:

(12)^2 = 144

Next, we multiply the result by the height:

144 \times 9 = 1296

Now, we multiply the result by $\frac{1}{3}$ and $\pi$ :

\frac{1}{3} \times 1296 \times \pi = 432 \pi

Conclusion

Therefore, the exact volume of a cone with a height of 9 and a radius of 12 is $432 \pi$ .

Comparison of Options

Let's compare our result with the given options:

A. $392 \pi$ B. $432 \pi$ C. $200 \pi$ D. $60 \pi$

Our result matches option B, which is $432 \pi$ .

Real-World Applications of the Volume of a Cone

The volume of a cone has numerous real-world applications, including:

Architecture: The volume of a cone is used to calculate the volume of a building's dome or a cone-shaped roof.
Engineering: The volume of a cone is used to calculate the volume of a cone-shaped tank or a cone-shaped pipe.
Physics: The volume of a cone is used to calculate the volume of a cone-shaped container or a cone-shaped object.

Tips and Tricks for Calculating the Volume of a Cone

Here are some tips and tricks for calculating the volume of a cone:

Use the formula: The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$ .
Plug in the values: Plug in the values of the radius and height into the formula.
Calculate the square of the radius: Calculate the square of the radius before multiplying it by the height.
Multiply the results: Multiply the results of the square of the radius and the height by $\frac{1}{3}$ and $\pi$ .

Common Mistakes to Avoid

Here are some common mistakes to avoid when calculating the volume of a cone:

Forgetting to square the radius: Forgetting to square the radius before multiplying it by the height.
Forgetting to multiply by $\frac{1}{3}$ and $\pi$ : Forgetting to multiply the result by $\frac{1}{3}$ and $\pi$ .
Using the wrong formula: Using the wrong formula for the volume of a cone.

Q&A

Q: What is the formula for the volume of a cone?

A: The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$ .

Q: How do I calculate the volume of a cone with a height of 9 and a radius of 12?

A: To calculate the volume of a cone with a height of 9 and a radius of 12, plug these values into the formula: $V = \frac{1}{3} \pi (12)^2 (9)$ .

Q: What is the square of the radius?

A: The square of the radius is $(12)^2 = 144$ .

Q: What is the result of multiplying the square of the radius by the height?

A: The result of multiplying the square of the radius by the height is $144 \times 9 = 1296$ .

Q: What is the result of multiplying the result by $\frac{1}{3}$ and $\pi$ ?

A: The result of multiplying the result by $\frac{1}{3}$ and $\pi$ is $\frac{1}{3} \times 1296 \times \pi = 432 \pi$ .

Q: What is the exact volume of a cone with a height of 9 and a radius of 12?

A: The exact volume of a cone with a height of 9 and a radius of 12 is $432 \pi$ .

Q: What are some real-world applications of the volume of a cone?

A: The volume of a cone has numerous real-world applications, including architecture, engineering, and physics.

Q: What are some tips and tricks for calculating the volume of a cone?

A: Some tips and tricks for calculating the volume of a cone include using the formula, plugging in the values, calculating the square of the radius, and multiplying the results by $\frac{1}{3}$ and $\pi$ .

Q: What are some common mistakes to avoid when calculating the volume of a cone?

A: Some common mistakes to avoid when calculating the volume of a cone include forgetting to square the radius, forgetting to multiply by $\frac{1}{3}$ and $\pi$ , and using the wrong formula.