The exact volume of a right cylinder with a base radius of 10 cm and a height of 6 cm is 600π cm³.
Understanding Cylinder Volume: The Basics
A right cylinder is a fundamental three-dimensional geometric shape characterized by two parallel and congruent circular bases connected by a curved surface. The axis connecting the centers of its bases is perpendicular to the bases. Calculating its volume is a core concept in geometry, with widespread utility in various real-world scenarios, from engineering to everyday tasks.
The Formula for Cylinder Volume
To determine the volume ($V$) of any right cylinder, you need two essential measurements: its base radius ($r$) and its height ($h$). The standard formula used is:
$V = \pi r^2 h$
Where:
- $V$ represents the total volume of the cylinder.
- $\pi$ (pi) is a mathematical constant, approximately 3.14159.
- $r$ is the radius of the circular base.
- $h$ is the perpendicular height of the cylinder between its two bases.
Calculating the Volume: A Step-by-Step Example
Let's calculate the volume for a right cylinder with the specific dimensions provided:
- Base Radius ($r$): 10 cm
- Height ($h$): 6 cm
Follow these steps to find the volume:
- Square the radius: Begin by squaring the base radius.
$r^2 = (10 \text{ cm})^2 = 100 \text{ cm}^2$ - Multiply by the height: Next, multiply the squared radius by the height of the cylinder.
$100 \text{ cm}^2 \times 6 \text{ cm} = 600 \text{ cm}^3$ - Multiply by $\pi$: Finally, multiply this result by $\pi$ to get the exact volume.
$V = 600\pi \text{ cm}^3$
Therefore, the exact volume of this specific right cylinder is 600π cubic centimeters. If an approximate numerical value is needed, using $\pi \approx 3.14159$ gives $V \approx 600 \times 3.14159 \approx 1884.954 \text{ cm}^3$.
Key Dimensions and Volume Summary
For clear understanding, here's a summary of the cylinder's properties:
| Property | Value | Unit |
|---|---|---|
| Base Radius | 10 | cm |
| Height | 6 | cm |
| Volume | $600\pi$ (exact) | cm$^3$ |
| Volume | $\approx 1884.95$ (approx.) | cm$^3$ |
Practical Applications of Cylinder Volume
The ability to calculate cylinder volume is essential across numerous disciplines and everyday scenarios:
- Engineering and Design: Used in designing pressure vessels, pipes, engine components, and hydraulic systems.
- Construction: Critical for estimating the amount of concrete needed for cylindrical pillars, foundations, or the capacity of water storage tanks.
- Manufacturing and Packaging: Determines the capacity of cans, bottles, and other cylindrical containers for products like food, beverages, and chemicals.
- Science and Research: Applied in laboratories to calculate the volume of liquids in graduated cylinders or the displacement of objects.
- Agriculture: Estimating the capacity of silos for grain storage or water tanks for irrigation.
Ensuring Accuracy in Measurements
Accurate measurement of both the radius and height is paramount for precise volume calculations. Minor errors in these initial measurements can lead to significant discrepancies in the final volume, impacting material costs, storage capacity, or the outcome of scientific experiments. For more in-depth information on cylinders and their properties, you can explore resources like Wikipedia's article on Cylinder (geometry).
