Calculating Cylinder Volume from Curved Surface Area and Circumference

Let's find the volume of a cylinder given its curved surface area and the circumference of its base. We are provided with the following information:

• Curved Surface Area (CSA) = 440 cm²
• Base Circumference (C) = 44 cm

We need to calculate the Volume (V) of the cylinder.

Formulas Needed for Cylinder Calculations

To solve this problem, we will use the standard formulas for a cylinder:

• Base Circumference, $C = 2\pi r$
• Curved Surface Area, $CSA = 2\pi rh$
• Volume, $V = \pi r^2 h$

Where 'r' is the radius of the base and 'h' is the height of the cylinder.

Step-by-Step Calculation of Cylinder Volume

We can find the volume by first determining the radius and height of the cylinder using the given information.

Step 1: Find the Radius (r) from the Base Circumference

We know the base circumference is 44 cm. Using the formula for circumference:

$C = 2\pi r$

$44 \text{ cm} = 2\pi r$

To find the radius 'r', we rearrange the formula:

$r = \frac{44}{2\pi} = \frac{22}{\pi}$

Using the approximation $\pi \approx \frac{22}{7}$:

$r = \frac{22}{(22/7)} = 22 \times \frac{7}{22} = 7 \text{ cm}$

The radius of the base is 7 cm.

Step 2: Find the Height (h) from the Curved Surface Area

The curved surface area is given as 440 cm². The formula for CSA is:

$CSA = 2\pi rh$

Notice that $2\pi r$ is the circumference, which we know is 44 cm. So, we can write the formula as:

$CSA = C \times h$

Substitute the known values:

$440 \text{ cm}^2 = 44 \text{ cm} \times h$

To find the height 'h', we rearrange the formula:

$h = \frac{440 \text{ cm}^2}{44 \text{ cm}} = 10 \text{ cm}$

The height of the cylinder is 10 cm.

Step 3: Calculate the Volume (V) of the Cylinder

Now that we have the radius (r = 7 cm) and the height (h = 10 cm), we can calculate the volume using the formula:

$V = \pi r^2 h$

Substitute the values of r and h, and use $\pi \approx \frac{22}{7}$:

$V = \pi (7 \text{ cm})^2 (10 \text{ cm})$

$V = \pi (49 \text{ cm}^2) (10 \text{ cm})$

$V = 490\pi \text{ cm}^3$

Using $\pi \approx \frac{22}{7}$:

$V = 490 \times \frac{22}{7} \text{ cm}^3$

$V = (70 \times 7) \times \frac{22}{7} \text{ cm}^3$

$V = 70 \times 22 \text{ cm}^3$

$V = 1540 \text{ cm}^3$

The volume of the cylinder is 1540 cm³.

Revision Table: Cylinder Formulas Summary

Additional Information on Cylinder Geometry

A cylinder is a 3D solid shape with two identical bases that are parallel circles and a curved surface connecting them. The line segment joining the centers of the circular bases is called the axis of the cylinder. If the axis is perpendicular to the bases, it's called a right circular cylinder, which is the type usually considered in these problems.

• Total Surface Area (TSA): This includes the area of the two bases plus the curved surface area. The area of one base is $\pi r^2$. So, TSA = $2\pi r^2 + 2\pi rh = 2\pi r(r+h)$.
• Units: It's important to keep track of units. Circumference is in linear units (cm), area is in square units (cm²), and volume is in cubic units (cm³).
• Relationship between Formulas: Notice how the formulas are interconnected. The CSA formula $2\pi rh$ can be seen as the circumference $2\pi r$ multiplied by the height $h$. The volume formula $\pi r^2 h$ is the base area $\pi r^2$ multiplied by the height $h$.

Understanding these formulas and their relationships helps in solving various problems involving cylinders.