The exact volume of a solid hemisphere with a radius of 7 cm is $\frac{686}{3}\pi$ cubic centimeters.
Understanding Hemisphere Volume
A hemisphere is fundamentally half of a sphere. To calculate its volume, we first need to recall the formula for the volume of a full sphere and then divide it by two. This geometric form is a common sight, from architectural domes to everyday objects.
The standard formula for the volume of a sphere is:
$V_{sphere} = \frac{4}{3}\pi r^3$
Consequently, the formula for the volume of a hemisphere is:
$V_{hemisphere} = \frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$
Detailed Calculation for a 7cm Radius Hemisphere
Let's apply this specific formula to a hemisphere that has a radius of 7 cm.
- Identify the Radius: The radius ($r$) of the hemisphere is given as 7 cm.
- Calculate the Cube of the Radius: As established, the cube of the radius ($r^3$) is $7^3 = 7 \times 7 \times 7 = 343$ cubic centimeters.
- Substitute Values into the Hemisphere Volume Formula:
$V{hemisphere} = \frac{2}{3} \pi (7 \text{ cm})^3$
$V{hemisphere} = \frac{2}{3} \pi (343 \text{ cm}^3)$
$V{hemisphere} = \frac{2 \times 343}{3}\pi \text{ cm}^3$
$V{hemisphere} = \frac{686}{3}\pi \text{ cm}^3$
This expression, $\frac{686}{3}\pi$ cubic centimeters, provides the exact volume of the hemisphere.
Exact vs. Approximate Volume
While the exact volume is expressed with $\pi$, an approximate numerical value is often used for practical applications. Using $\pi \approx 3.14159$, the approximate volume can be calculated as:
$V_{hemisphere} \approx \frac{686}{3} \times 3.14159 \text{ cm}^3 \approx 228.666... \times 3.14159 \text{ cm}^3 \approx 718.66 \text{ cm}^3$.
This aligns with the provided approximate value of 718.66 cubic cm. It's crucial to understand the distinction between an exact answer (which includes $\pi$) and its numerical approximation.
Key Hemisphere Properties and Volumes
Here is a summary of the key values related to this hemisphere:
| Property | Value | Unit |
|---|---|---|
| Radius ($r$) | 7 | cm |
| Radius cubed ($r^3$) | 343 | $cm^3$ |
| Exact Volume | $\frac{686}{3}\pi$ | $cm^3$ |
| Approximate Volume | 718.66 | $cm^3$ |
Practical Applications of Hemisphere Volume Calculations
Calculating the volume of a hemisphere has diverse applications across various disciplines:
- Engineering and Design: Engineers often need to determine the capacity of hemispherical tanks, domes, or pressure vessels to ensure they meet specific requirements.
- Architecture: Architects use these calculations to plan the internal space and material requirements for domed structures, optimizing both aesthetics and functionality.
- Manufacturing: In manufacturing, knowing the volume helps in estimating the amount of material needed for casting or molding hemispherical components, minimizing waste.
- Science and Research: Scientists might use volume calculations in fields like fluid dynamics or to analyze the properties of spherical or hemispherical objects in experiments.
For more detailed explanations and examples of volume calculations for various 3D shapes, you can explore resources such as Khan Academy's geometry section on volumes.
