Surface To Volume Ratio Of Hexakis Octahedron Given Midsphere Radius Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{12\sqrt{543+176\sqrt{2}}}{\sqrt{6(986+607\sqrt{2})}} \times \frac{1+2\sqrt{2}}{4 \times \text{Midsphere Radius}} \]

Surface to Volume Ratio:

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1. What is Surface to Volume Ratio of Hexakis Octahedron?

The Surface to Volume Ratio of a Hexakis Octahedron is a geometric property that represents the relationship between the total surface area and the total volume of this complex polyhedron. It's an important parameter in materials science and geometry studies.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ \text{Surface to Volume Ratio} = \frac{12\sqrt{543+176\sqrt{2}}}{\sqrt{6(986+607\sqrt{2})}} \times \frac{1+2\sqrt{2}}{4 \times \text{Midsphere Radius}} \]

Where:

Midsphere Radius - The radius of the sphere that touches all edges of the Hexakis Octahedron
\(\sqrt{2}\) - Square root of 2 (approximately 1.4142)

Explanation: The formula combines geometric constants specific to the Hexakis Octahedron with the given midsphere radius to calculate the surface to volume ratio.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is crucial in various applications including material science, chemical reactivity studies, heat transfer calculations, and understanding geometric properties of complex polyhedra.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and greater than zero. The calculator will compute the corresponding surface to volume ratio.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Octahedron?
A: A Hexakis Octahedron is a Catalan solid that is the dual of the truncated cuboctahedron, featuring 48 faces, 72 edges, and 26 vertices.

Q2: What is the significance of the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron, providing important geometric information about the shape's dimensions.

Q3: What units are used in this calculation?
A: The midsphere radius is input in meters, and the surface to volume ratio is output in reciprocal meters (m⁻¹).

Q4: Are there any limitations to this formula?
A: This formula is specifically designed for perfect Hexakis Octahedrons and assumes ideal geometric conditions.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for the given formula, though practical measurements of the midsphere radius may introduce measurement errors.