Partially Filled Sphere Calculator

What is a Partially Filled Sphere?

A partially filled sphere is a spherical container that is not completely full of liquid or material. It's commonly used in various industries for storage tanks, pressure vessels, and scientific applications. Understanding the volume and properties of a partially filled sphere is crucial for accurate measurements and efficient use of resources.

How to Calculate Partially Filled Sphere Properties

To fully understand a partially filled sphere, we need to calculate several key properties: its radius, fill height, total volume, filled volume, and fill percentage. Each of these properties provides unique information about the sphere's dimensions and capacity.

Formulas

Here are the essential formulas for a partially filled sphere:

1. Total Volume of Sphere (V_total):

\[ V_{total} = \frac{4}{3}\pi r^3 \]

2. Volume of Filled Portion (V_filled):

\[ V_{filled} = \frac{\pi h^2(3r - h)}{3} \]

3. Fill Percentage:

\[ Fill\% = \frac{V_{filled}}{V_{total}} \times 100\% \]

Where:

• \(r\) is the radius of the sphere
• \(h\) is the fill height (measured from the bottom of the sphere)
• \(\pi\) is approximately 3.14159

Calculation Steps

1. Determine the radius (r) of the sphere and the fill height (h)
2. Calculate the total volume of the sphere using \(V_{total} = \frac{4}{3}\pi r^3\)
3. Calculate the filled volume using \(V_{filled} = \frac{\pi h^2(3r - h)}{3}\)
4. Calculate the fill percentage using \(Fill\% = \frac{V_{filled}}{V_{total}} \times 100\%\)

Example and Visual Representation

Let's calculate the properties of a partially filled sphere with a radius of 2 meters and a fill height of 1.5 meters:

1. Given: \(r = 2\) m, \(h = 1.5\) m
2. Total Volume: \(V_{total} = \frac{4}{3}\pi 2^3 \approx 33.51\) cubic meters
3. Filled Volume: \(V_{filled} = \frac{\pi 1.5^2(3 \times 2 - 1.5)}{3} \approx 11.78\) cubic meters
4. Fill Percentage: \(Fill\% = \frac{11.78}{33.51} \times 100\% \approx 35.15\%\)

Here's a visual representation of this partially filled sphere:

In this diagram, you can see a 2D representation of our partially filled sphere with radius 2 meters and fill height 1.5 meters. The red line shows the radius, and the green line shows the fill height. The total volume (V_total), filled volume (V_filled), and fill percentage are labeled.