Formula for the Volume of a Cone

Oct 13, 2025 by magicalceylontours.com 49 views

Introduction

In this problem, we are given a cone with a height of 14 cm and a volume of $$132 , \text{cm}^3$$. We are asked to find the radius of the circular base of the cone. To solve this problem, we will use the formula for the volume of a cone, which is given by:

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

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

Formula for the Volume of a Cone

The formula for the volume of a cone is given by:

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

This formula can be derived by considering the cone as a stack of disks, each with a radius of $r$ and a thickness of $h$ . The volume of each disk is given by:

\pi r^2 h

Since there are $h$ disks in the cone, the total volume is given by:

V = h \cdot \pi r^2

However, since the disks are stacked, we need to divide the total volume by 3 to get the correct formula:

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

Substituting the Given Values

We are given that the volume of the cone is $$132 , \text{cm}^3$$ and the height is 14 cm. We can substitute these values into the formula for the volume of a cone:

132 = \frac{1}{3} \cdot \frac{22}{7} \cdot r^2 \cdot 14

Simplifying the Equation

We can simplify the equation by canceling out the common factors:

132 = \frac{22}{7} \cdot r^2 \cdot 4

Solving for the Radius

We can solve for the radius by dividing both sides of the equation by $$\frac{22}{7} \cdot 4$$:

r^2 = \frac{132 \cdot 7}{22 \cdot 4}

r^2 = \frac{924}{88}

r^2 = 10.5

Taking the Square Root

We can take the square root of both sides of the equation to find the radius:

r = \sqrt{10.5}

r = 3.24

Conclusion

In this problem, we used the formula for the volume of a cone to find the radius of the circular base. We substituted the given values into the formula and simplified the equation to solve for the radius. The final answer is $$r = 3.24$$ cm.

Discussion

This problem is a classic example of how to use the formula for the volume of a cone to find the radius of the circular base. The formula is a fundamental concept in mathematics and is used in a wide range of applications, including physics, engineering, and computer science. The problem requires the student to apply the formula and simplify the equation to find the solution, which is an important skill in mathematics.

Real-World Applications

The formula for the volume of a cone has many real-world applications, including:

Physics: The formula is used to calculate the volume of a cone-shaped object, such as a cone-shaped container or a cone-shaped projectile.
Engineering: The formula is used to design and optimize cone-shaped structures, such as cone-shaped tanks or cone-shaped buildings.
Computer Science: The formula is used in computer graphics to create 3D models of cone-shaped objects.

Conclusion

In conclusion, the problem of finding the radius of a cone with a given volume and height is a classic example of how to use the formula for the volume of a cone. The formula is a fundamental concept in mathematics and has many real-world applications. The problem requires the student to apply the formula and simplify the equation to find the solution, which is an important skill in mathematics.

Introduction

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

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

Formula for the Volume of a Cone

The formula for the volume of a cone is given by:

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

This formula can be derived by considering the cone as a stack of disks, each with a radius of $r$ and a thickness of $h$ . The volume of each disk is given by:

\pi r^2 h

Since there are $h$ disks in the cone, the total volume is given by:

V = h \cdot \pi r^2

However, since the disks are stacked, we need to divide the total volume by 3 to get the correct formula:

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

Substituting the Given Values

We are given that the volume of the cone is $$132 , \text{cm}^3$$ and the height is 14 cm. We can substitute these values into the formula for the volume of a cone:

132 = \frac{1}{3} \cdot \frac{22}{7} \cdot r^2 \cdot 14

Simplifying the Equation

We can simplify the equation by canceling out the common factors:

132 = \frac{22}{7} \cdot r^2 \cdot 4

Solving for the Radius

We can solve for the radius by dividing both sides of the equation by $$\frac{22}{7} \cdot 4$$:

r^2 = \frac{132 \cdot 7}{22 \cdot 4}

r^2 = \frac{924}{88}

r^2 = 10.5

Taking the Square Root

We can take the square root of both sides of the equation to find the radius:

r = \sqrt{10.5}

r = 3.24

Conclusion

Discussion

Real-World Applications

The formula for the volume of a cone has many real-world applications, including:

Physics: The formula is used to calculate the volume of a cone-shaped object, such as a cone-shaped container or a cone-shaped projectile.
Engineering: The formula is used to design and optimize cone-shaped structures, such as cone-shaped tanks or cone-shaped buildings.
Computer Science: The formula is used in computer graphics to create 3D models of cone-shaped objects.

Q&A

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

A: The formula for the volume of a cone is given by:

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

Q: What is the radius of the circular base of the cone?

A: The radius of the circular base of the cone is $$r = 3.24$$ cm.

Q: How do we find the radius of the circular base of the cone?

A: We find the radius of the circular base of the cone by substituting the given values into the formula for the volume of a cone and simplifying the equation.

Q: What is the height of the cone?

A: The height of the cone is 14 cm.

Q: What is the volume of the cone?

A: The volume of the cone is $$132 , \text{cm}^3$$.

Q: What is the formula for the volume of a cone in terms of the radius and height?

A: The formula for the volume of a cone in terms of the radius and height is given by:

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

Q: How do we simplify the equation to find the radius of the circular base of the cone?

A: We simplify the equation by canceling out the common factors and then solving for the radius.

Q: What is the final answer to the problem?

A: The final answer to the problem is $$r = 3.24$$ cm.

Conclusion

In this article, we have discussed the problem of finding the radius of a cone with a given volume and height. We have used the formula for the volume of a cone to find the radius of the circular base. We have also provided a Q&A section to answer common questions related to the problem. The final answer to the problem is $$r = 3.24$$ cm.