The Volume of the Cylinder

Oct 13, 2025 by miodeya.com 26 views

Introduction

In mathematics, a cylinder is a three-dimensional shape with two parallel and circular bases connected by a curved lateral surface. The volume of a cylinder is a crucial property that can be calculated using the formula: $$\pi r^2 h$$, where $$r$$ is the radius and $$h$$ is the height of the cylinder. However, in this article, we will explore a cylinder with a base diameter of $$x$$ units and a volume of $$\pi x^3$$ cubic units. We will examine the properties of this cylinder and determine which statements about it are true.

The Volume of the Cylinder

The volume of the cylinder is given as $$\pi x^3$$ cubic units. To understand this, we need to recall the formula for the volume of a cylinder: $$\pi r^2 h$$. Since the base diameter is $$x$$ units, the radius is half of the diameter, which is $$\frac{x}{2}$$ units. Substituting this value into the volume formula, we get:

\pi \left(\frac{x}{2}\right)^2 h = \pi \frac{x^2}{4} h

However, the given volume is $$\pi x^3$$ cubic units. This suggests that the height of the cylinder is not a constant value, but rather a function of the radius. To determine the relationship between the radius and the height, we can equate the two volume expressions:

\pi \frac{x^2}{4} h = \pi x^3

Simplifying this equation, we get:

\frac{x^2}{4} h = x^3

Dividing both sides by $$x^2$$, we obtain:

\frac{h}{4} = x

Multiplying both sides by $$4$$, we find:

h = 4x

This result indicates that the height of the cylinder is $$4x$$ units.

The Radius of the Cylinder

Now that we have determined the height of the cylinder, we can examine the radius. The radius is half of the diameter, which is $$\frac{x}{2}$$ units. This is a direct result of the definition of the diameter and the radius.

The Area of the Base

The area of the base of the cylinder can be calculated using the formula: $$\pi r^2$$. Substituting the value of the radius, we get:

\pi \left(\frac{x}{2}\right)^2 = \pi \frac{x^2}{4}

This is the area of the base of the cylinder.

Conclusion

In conclusion, we have examined a cylinder with a base diameter of $$x$$ units and a volume of $$\pi x^3$$ cubic units. We have determined that the height of the cylinder is $$4x$$ units and the radius is $$\frac{x}{2}$$ units. We have also calculated the area of the base of the cylinder, which is $$\pi \frac{x^2}{4}$$ square units.

Final Answer

Based on the analysis, we can conclude that the following statements about the cylinder are true:

The radius of the cylinder is $$\frac{x}{2}$$ units.
The area of the base of the cylinder is $$\pi \frac{x^2}{4}$$ square units.

The other options are incorrect, as the radius is not $$2x$$ units, and the area of the base is not $$\pi x^2$$ square units.

Introduction

In our previous article, we explored a cylinder with a base diameter of $$x$$ units and a volume of $$\pi x^3$$ cubic units. We determined that the height of the cylinder is $$4x$$ units and the radius is $$\frac{x}{2}$$ units. We also calculated the area of the base of the cylinder, which is $$\pi \frac{x^2}{4}$$ square units. In this article, we will answer some frequently asked questions about the cylinder.

Q&A

Q: What is the formula for the volume of the cylinder?

A: The formula for the volume of the cylinder is $$\pi r^2 h$$, where $$r$$ is the radius and $$h$$ is the height of the cylinder.

Q: How do we calculate the radius of the cylinder?

A: The radius of the cylinder is half of the diameter, which is $$\frac{x}{2}$$ units.

Q: What is the relationship between the radius and the height of the cylinder?

A: The height of the cylinder is $$4x$$ units, and the radius is $$\frac{x}{2}$$ units. This indicates that the height is four times the radius.

Q: How do we calculate the area of the base of the cylinder?

A: The area of the base of the cylinder can be calculated using the formula: $$\pi r^2$$. Substituting the value of the radius, we get:

\pi \left(\frac{x}{2}\right)^2 = \pi \frac{x^2}{4}

This is the area of the base of the cylinder.

Q: What is the volume of the cylinder in terms of the radius?

A: The volume of the cylinder is $$\pi r^2 h$$, where $$r$$ is the radius and $$h$$ is the height of the cylinder. Substituting the value of the height, we get:

\pi r^2 (4r) = 4 \pi r^3

This is the volume of the cylinder in terms of the radius.

Q: What is the relationship between the diameter and the radius of the cylinder?

A: The diameter of the cylinder is $$x$$ units, and the radius is half of the diameter, which is $$\frac{x}{2}$$ units.

Q: How do we calculate the circumference of the base of the cylinder?

A: The circumference of the base of the cylinder can be calculated using the formula: $$2 \pi r$$. Substituting the value of the radius, we get:

2 \pi \left(\frac{x}{2}\right) = \pi x

This is the circumference of the base of the cylinder.

Conclusion

In conclusion, we have answered some frequently asked questions about the cylinder with a base diameter of $$x$$ units and a volume of $$\pi x^3$$ cubic units. We have determined that the height of the cylinder is $$4x$$ units and the radius is $$\frac{x}{2}$$ units. We have also calculated the area of the base of the cylinder, which is $$\pi \frac{x^2}{4}$$ square units.

Final Answer

Based on the analysis, we can conclude that the following statements about the cylinder are true:

The radius of the cylinder is $$\frac{x}{2}$$ units.
The area of the base of the cylinder is $$\pi \frac{x^2}{4}$$ square units.
The height of the cylinder is $$4x$$ units.
The circumference of the base of the cylinder is $$\pi x$$ units.

The other options are incorrect, as the radius is not $$2x$$ units, and the area of the base is not $$\pi x^2$$ square units.