Volume Of Ellipsoid Given Surface Area, Second And Third Semi Axes Calculator

Formula Used:

\[ V = \frac{4\pi}{3} \times b \times c \times \left( \frac{3 \times (SA - 2\pi \times b \times c)}{4\pi \times \left( \frac{b}{c} + \frac{c}{b} + 1 \right)} \right)^{\frac{1}{2}} \]

Volume (V):

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1. What is Volume Of Ellipsoid?

The volume of an ellipsoid represents the three-dimensional space enclosed by its surface. An ellipsoid is a quadric surface that generalizes an ellipse to three dimensions, defined by three semi-axes (a, b, c).

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{4\pi}{3} \times b \times c \times \left( \frac{3 \times (SA - 2\pi \times b \times c)}{4\pi \times \left( \frac{b}{c} + \frac{c}{b} + 1 \right)} \right)^{\frac{1}{2}} \]

Where:

\( V \) — Volume of the ellipsoid (m³)
\( SA \) — Surface area of the ellipsoid (m²)
\( b \) — Second semi-axis (m)
\( c \) — Third semi-axis (m)
\( \pi \) — Mathematical constant pi (≈3.1416)

Explanation: This formula calculates the volume of an ellipsoid when given its surface area and two of its three semi-axes, solving for the third semi-axis using the surface area formula.

3. Importance of Volume Calculation

Details: Calculating the volume of ellipsoids is crucial in various fields including physics, engineering, astronomy, and medical imaging where ellipsoidal shapes are commonly encountered.

4. Using the Calculator

Tips: Enter the surface area in square meters, and both semi-axes in meters. All values must be positive numbers. The calculator will compute the volume in cubic meters.

5. Frequently Asked Questions (FAQ)

Q1: What is an ellipsoid?
A: An ellipsoid is a three-dimensional geometric shape that is the analog of an ellipse in three dimensions, defined by three perpendicular axes of different lengths.

Q2: How is this different from a sphere?
A: A sphere is a special case of an ellipsoid where all three semi-axes are equal. In an ellipsoid, the semi-axes can have different lengths.

Q3: What are practical applications of ellipsoid volume calculations?
A: Used in calculating volumes of planets, storage tanks, sports balls (rugby, American football), and in medical imaging for tumor volume measurements.

Q4: Are there limitations to this calculation?
A: The formula assumes a perfect ellipsoidal shape and may not be accurate for irregular shapes. It also requires precise measurement of surface area and semi-axes.

Q5: Can this formula be used for prolate and oblate spheroids?
A: Yes, both prolate (one axis longer than the other two equal axes) and oblate (one axis shorter than the other two equal axes) spheroids are special cases of ellipsoids.