Finding Cylinder Base Radius from Rolled Rectangle

This question asks us to determine the base radius of a cylinder formed by rolling a rectangular sheet of paper. We are given the dimensions of the rectangular sheet and told how it is rolled.

Understanding the Rolling Process

When a rectangular sheet of paper is rolled to form a cylinder, the dimensions of the rectangle relate directly to the dimensions of the cylinder:

• One dimension of the rectangle becomes the height of the cylinder.
• The other dimension of the rectangle becomes the circumference of the base circle of the cylinder.

In this specific problem:

• The rectangular sheet has a length of 6 cm and a width of 3 cm.
• The sheet is rolled such that the height of the cylinder is equal to the width of the paper.

This means:

• Height of the cylinder = Width of the paper = 3 cm.
• The length of the paper becomes the circumference of the base of the cylinder.

So, the circumference of the base of the cylinder is 6 cm.

Calculating the Base Radius

The formula for the circumference of a circle is given by \(C = 2 \pi r\), where \(C\) is the circumference and \(r\) is the radius of the circle. In this case, the circumference \(C\) is 6 cm.

We can set up the equation using this information:

\(2 \pi r = 6 \text{ cm}\)

To find the radius \(r\), we need to isolate \(r\) in the equation. We can do this by dividing both sides of the equation by \(2 \pi\):

\(r = \frac{6}{2 \pi} \text{ cm}\)

Now, we can simplify the fraction by dividing the numerator and the denominator by 2:

\(r = \frac{3}{\pi} \text{ cm}\)

Therefore, the base radius of the cylinder is \(\frac{3}{\pi}\) cm.

Summary of Dimensions

Final Answer Derivation

Using the relationship between the rectangle's length and the cylinder's circumference:

Circumference = Length of rectangle

\(2 \pi r = 6\)

Solve for \(r\):

\(r = \frac{6}{2 \pi}\)

\(r = \frac{3}{\pi}\)

The base radius of the cylinder is \(\frac{3}{\pi}\) cm.

Revision Table - Cylinder Dimensions

Additional Information - Rolling Rectangles into Cylinders

Understanding how a 2D shape like a rectangle transforms into a 3D shape like a cylinder is a fundamental concept in geometry. When you roll a rectangle, there are two possibilities for how the cylinder is formed, depending on which dimension becomes the height:

1. Rolling along the Length: If you roll the rectangle so that the sides of length become the height, then the width of the rectangle becomes the circumference of the base. The height of the cylinder is the length of the rectangle.
2. Rolling along the Width: If you roll the rectangle so that the sides of width become the height (as in this problem), then the length of the rectangle becomes the circumference of the base. The height of the cylinder is the width of the rectangle.

It's important to read the question carefully to determine which dimension of the rectangle corresponds to the height of the resulting cylinder. This determines which dimension becomes the circumference.