A square sheet of side length 44 cm is rolled along one of its sides to form a cylinder by making opposite edges just to touch each other. What is the volume of the cylinder ? (Take π = 22/7)
6776 cubic cm
Calculating Volume of a Cylinder from a Square Sheet
This problem involves visualizing how a two-dimensional shape (a square sheet) is transformed into a three-dimensional shape (a cylinder) and then calculating its volume.
Understanding the Problem Transformation
We are given a square sheet with a side length of 44 cm. When this sheet is rolled along one of its sides so that opposite edges touch, it forms a cylinder. Let's see how the dimensions of the square relate to the dimensions of the resulting cylinder:
- One side of the square becomes the height of the cylinder.
- The other side of the square becomes the circumference of the base circle of the cylinder.
In this case, the side length of the square is 44 cm. So:
- Height of the cylinder (h) = 44 cm
- Circumference of the base of the cylinder (C) = 44 cm
We need to find the volume of this cylinder. The formula for the volume of a cylinder is \(V = \pi r^2 h\), where \(r\) is the radius of the base and \(h\) is the height. We are given \(h = 44\) cm and we need to find \(r\). We can find \(r\) using the circumference.
Finding the Radius of the Cylinder Base
The circumference of a circle is given by the formula \(C = 2\pi r\). We know the circumference is 44 cm and we are asked to use \( \pi = 22/7 \). Let's plug these values into the formula to find the radius \(r\).
Given:
- Circumference \(C = 44\) cm
- \( \pi = 22/7 \)
Formula:
\( C = 2\pi r \)
Substitute the given values:
\( 44 = 2 \times \left(\frac{22}{7}\right) \times r \)
\( 44 = \frac{44}{7} \times r \)
Now, solve for \(r\):
\( r = 44 \times \frac{7}{44} \)
\( r = 7 \) cm
So, the radius of the base of the cylinder is 7 cm.
Calculating the Volume of the Cylinder
Now that we have the radius \(r = 7\) cm and the height \(h = 44\) cm, we can calculate the volume of the cylinder using the formula \(V = \pi r^2 h\). We will use \( \pi = 22/7 \).
Given:
- Radius \(r = 7\) cm
- Height \(h = 44\) cm
- \( \pi = 22/7 \)
Formula:
\( V = \pi r^2 h \)
Substitute the values:
\( V = \frac{22}{7} \times (7)^2 \times 44 \)
\( V = \frac{22}{7} \times 49 \times 44 \)
We can cancel out the 7 in the denominator with one of the 7s from \(49 = 7 \times 7\):
\( V = 22 \times \frac{49}{7} \times 44 \)
\( V = 22 \times 7 \times 44 \)
Now, perform the multiplication:
\( V = 154 \times 44 \)
Let's do the multiplication:
154 x 44 ----- 616 (154 * 4) 6160 (154 * 40) ----- 6776
\( V = 6776 \) cubic cm
The volume of the cylinder is 6776 cubic cm.
Summary of Calculations
| Dimension/Property | Value | Calculation Method |
|---|---|---|
| Square Side Length | 44 cm | Given |
| Cylinder Height (h) | 44 cm | Side length used for rolling |
| Cylinder Base Circumference (C) | 44 cm | Other side length of the square |
| Cylinder Base Radius (r) | 7 cm | \( C = 2\pi r \Rightarrow 44 = 2 \times \frac{22}{7} \times r \) |
| Cylinder Volume (V) | 6776 cubic cm | \( V = \pi r^2 h = \frac{22}{7} \times 7^2 \times 44 \) |
The calculated volume of the cylinder is 6776 cubic cm.
Revision Table: Geometry Formulas
| Shape | Property | Formula |
|---|---|---|
| Square | Side Length | s |
| Circle | Circumference (C) | \( 2\pi r \) |
| Circle | Area (A) | \( \pi r^2 \) |
| Cylinder | Volume (V) | \( \pi r^2 h \) |
| Cylinder | Curved Surface Area | \( 2\pi r h \) |
| Cylinder | Total Surface Area | \( 2\pi r (r + h) \) |
Additional Information: Rolling Shapes
When a 2D shape like a rectangle or a square is rolled to form a 3D shape like a cylinder:
- One dimension of the 2D shape becomes the height of the cylinder.
- The other dimension of the 2D shape becomes the circumference of the cylinder's base.
- If you roll a rectangle along its length, the length becomes the circumference and the width becomes the height.
- If you roll a rectangle along its width, the width becomes the circumference and the length becomes the height.
- In the case of a square, both sides are equal. The side along which it's rolled determines which side becomes the height and which becomes the circumference. The problem states it's rolled along "one of its sides," meaning one side forms the height and the other equal side forms the circumference.
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