The volume of a right circular cylinder with a curved surface area of 660 cm² and a base circumference of 66 cm is 3465 cm³.
Calculating the volume of a cylinder involves determining its radius and height first, using the provided surface area and circumference. This process utilizes fundamental geometric formulas.
Understanding Cylinder Geometry
A right circular cylinder is a three-dimensional shape with two parallel circular bases and a curved surface connecting them. To find its volume, we need two key measurements: the radius of its base and its height.
Key Formulas for a Right Circular Cylinder:
- Circumference of the Base (C): $C = 2\pi r$
- Where $r$ is the radius of the base.
- Curved Surface Area (CSA): $CSA = 2\pi r h$
- Where $r$ is the radius of the base and $h$ is the height of the cylinder.
- Volume (V): $V = \pi r^2 h$
- Where $r$ is the radius of the base and $h$ is the height of the cylinder.
For these calculations, we'll use the approximation of $\pi \approx \frac{22}{7}$ for precise results often expected in such problems.
Step-by-Step Calculation
Let's break down the calculation into clear, manageable steps.
1. Determine the Radius of the Base
Given the circumference of the base ($C = 66 \text{ cm}$), we can find the radius ($r$).
- Formula: $C = 2\pi r$
- Substitute values: $66 = 2 \times \frac{22}{7} \times r$
- Simplify: $66 = \frac{44}{7} r$
- Solve for r: $r = \frac{66 \times 7}{44} = \frac{3 \times 7}{2} = \frac{21}{2} = 10.5 \text{ cm}$
Thus, the radius of the cylinder's base is 10.5 cm.
2. Calculate the Height of the Cylinder
Next, we use the curved surface area ($CSA = 660 \text{ cm}^2$) and the calculated radius to find the height ($h$).
- Formula: $CSA = 2\pi r h$
- Recognize: We already know $2\pi r$ is the circumference, which is $66 \text{ cm}$.
- Substitute values: $660 = 66 \times h$
- Solve for h: $h = \frac{660}{66} = 10 \text{ cm}$
Therefore, the height of the cylinder is 10 cm.
3. Compute the Volume of the Cylinder
With both the radius and height determined, we can now calculate the volume ($V$).
- Formula: $V = \pi r^2 h$
- Substitute values: $V = \frac{22}{7} \times (10.5)^2 \times 10$
- Calculate $r^2$: $(10.5)^2 = 110.25$
- Substitute and multiply: $V = \frac{22}{7} \times 110.25 \times 10$
- Simplify: $V = \frac{22}{7} \times 1102.5$
- Final Calculation: $V = 22 \times 157.5 = 3465 \text{ cm}^3$
The volume of the right circular cylinder is 3465 cm³.
Summary of Dimensions and Volume
To make it easy to follow, here's a summary of the calculated dimensions and the final volume:
| Measurement | Value | Unit |
|---|---|---|
| Circumference of Base | 66 | cm |
| Curved Surface Area | 660 | cm² |
| Radius (r) | 10.5 | cm |
| Height (h) | 10 | cm |
| Volume (V) | 3465 | cm³ |
Understanding these steps allows for the accurate calculation of cylinder volumes given various initial parameters. For more details on cylinder properties, you can refer to resources like Wikipedia's Cylinder article.
