The exact volume of water in a cylinder 10cm high with a radius of 5cm, utilizing the approximation of pi ($\pi$) as 22/7, is 5500/7 cubic centimeters. This value is approximately 785.71 Cu.,cm.
Understanding Cylinder Volume Calculation
The volume of a cylinder quantifies the amount of space it occupies, or in this context, the capacity of water it can hold. To accurately determine this, two key dimensions are essential: the cylinder's radius and its height.
The Fundamental Formula for Volume
The standard mathematical formula used to calculate the volume ($V$) of any cylinder is:
$V = \pi r^2 h$
Where:
- $V$ represents the volume
- $\pi$ (pi) is a mathematical constant, approximately 3.14159
- $r$ is the radius of the cylinder's base
- $h$ is the height of the cylinder
For many practical applications and specific problem sets, $\pi$ is often approximated as $\frac{22}{7}$ to facilitate calculations.
Calculating the Volume of Your Cylinder
Given the specifications for your cylinder:
- Radius (r): 5 cm
- Height (h): 10 cm
- Approximation for Pi ($\pi$): $\frac{22}{7}$
We can substitute these values into the volume formula:
- Square the radius: $r^2 = (5 \text{ cm})^2 = 25 \text{ cm}^2$
- Multiply by height: $25 \text{ cm}^2 \times 10 \text{ cm} = 250 \text{ cm}^3$
- Multiply by Pi approximation: $V = \frac{22}{7} \times 250 \text{ cm}^3 = \frac{5500}{7} \text{ cm}^3$
Therefore, the exact volume of water in the cylinder, derived from using $\pi = \frac{22}{7}$, is $\frac{5500}{7}$ cubic centimeters.
Decimal Approximation for Practical Use
While $\frac{5500}{7} \text{ cm}^3$ is the exact fractional answer, it can be converted to a decimal for easier understanding and practical reporting.
$\frac{5500}{7} \text{ cm}^3 \approx 785.7142857... \text{ cm}^3$
This value is commonly rounded to two decimal places, making the approximate volume 785.71 Cu.,cm. This rounding often occurs in various measurements and reports for convenience.
Summary of Cylinder Volume Calculation
| Measurement | Value | Unit |
|---|---|---|
| Radius (r) | 5 | cm |
| Height (h) | 10 | cm |
| Pi ($\pi$) Used | 22/7 | - |
| Exact Volume | 5500/7 | cm³ |
| Approximate Volume | 785.71 | cm³ |
Additional Insights on Volume Measurement
- Precision Matters: The choice of $\pi$ approximation impacts the precision of the final volume. Using a more precise value for $\pi$ (e.g., from a calculator's built-in function, which is $\approx 3.14159265$) would yield a slightly different result ($3.14159265 \times 250 \approx 785.398 \text{ cm}^3$).
- Units of Volume: Volume is typically measured in cubic units, such as cubic centimeters (cm³), cubic meters (m³), or cubic inches (in³). For liquids, 1 cubic centimeter is equivalent to 1 milliliter (mL), which is useful for understanding the capacity of containers like this cylinder. For more information on volume, refer to resources like Wikipedia's entry on Volume.
- Real-world Applications: Calculating cylinder volume is fundamental in many fields, including engineering (for designing pipes, tanks), chemistry (for measuring liquid reagents), and even in daily life (for understanding the capacity of everyday containers).
