Volume Of Hexakis Icosahedron Given Surface To Volume Ratio Calculator

Formula Used:

\[ V = \frac{25}{88} \times \left( \frac{\frac{6}{5} \times \sqrt{10 \times (417 + 107 \times \sqrt{5})}}{R_{A/V} \times \sqrt{6 \times (185 + 82 \times \sqrt{5})}} \right)^3 \times \sqrt{6 \times (185 + 82 \times \sqrt{5})} \]

Volume of Hexakis Icosahedron (V):

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1. What is Volume of Hexakis Icosahedron?

The volume of a Hexakis Icosahedron is the quantity of three dimensional space enclosed by the entire surface of this complex polyhedron. It is an important geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( V \) — Volume of Hexakis Icosahedron (m³)
\( R_{A/V} \) — Surface to Volume Ratio (m⁻¹)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the volume based on the surface to volume ratio, incorporating mathematical constants and square root functions specific to the geometry of Hexakis Icosahedron.

3. Importance of Volume Calculation

Details: Calculating the volume of complex polyhedra like Hexakis Icosahedron is crucial for geometric analysis, material science applications, and understanding spatial properties in three-dimensional geometry.

4. Using the Calculator

Tips: Enter the surface to volume ratio in m⁻¹. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Icosahedron?
A: A Hexakis Icosahedron is a Catalan solid that is the dual of the truncated dodecahedron, featuring 120 faces, 180 edges, and 62 vertices.

Q2: What units should I use for input?
A: The surface to volume ratio should be entered in reciprocal meters (m⁻¹), and the output volume will be in cubic meters (m³).

Q3: Can this calculator handle very small values?
A: Yes, the calculator can handle values as small as 0.0001 m⁻¹, but extremely small values may approach computational limits.

Q4: What are typical surface to volume ratio values?
A: The surface to volume ratio depends on the specific dimensions of the Hexakis Icosahedron and can vary widely based on its size and proportions.

Q5: Is this formula exact or approximate?
A: This is an exact mathematical formula derived from the geometric properties of the Hexakis Icosahedron, though computational results may have minor rounding differences.