Midsphere Radius of Hexakis Octahedron given Total Surface Area Calculator

Formula Used:

\[ r_m = \frac{1 + 2\sqrt{2}}{4} \times \sqrt{\frac{7 \times TSA}{3 \times \sqrt{543 + 176\sqrt{2}}}} \]

Midsphere Radius of Hexakis Octahedron:

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1. What is the Midsphere Radius of Hexakis Octahedron?

The Midsphere Radius of a Hexakis Octahedron is defined as the radius of the sphere for which all the edges of the Hexakis Octahedron become a tangent line on that sphere. It is an important geometric property of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{1 + 2\sqrt{2}}{4} \times \sqrt{\frac{7 \times TSA}{3 \times \sqrt{543 + 176\sqrt{2}}}} \]

Where:

\( r_m \) — Midsphere Radius of Hexakis Octahedron (m)
\( TSA \) — Total Surface Area of Hexakis Octahedron (m²)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula calculates the midsphere radius based on the total surface area of the Hexakis Octahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties of the Hexakis Octahedron. It helps in various applications including crystallography, molecular modeling, and computer graphics.

4. Using the Calculator

Tips: Enter the total surface area of the Hexakis Octahedron in square meters. The value must be positive and greater than zero for accurate calculation.

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. It has 48 faces, 72 edges, and 26 vertices.

Q2: How is the midsphere radius different from the insphere radius?
A: The midsphere radius is the radius of the sphere tangent to all edges, while the insphere radius is tangent to all faces of the polyhedron.

Q3: What are typical values for midsphere radius?
A: The midsphere radius depends on the size of the Hexakis Octahedron. For a polyhedron with total surface area of 1 m², the midsphere radius is approximately 0.15-0.20 m.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is only applicable to the Hexakis Octahedron. Other polyhedra have different formulas for calculating their midsphere radii.

Q5: What is the significance of the constants in the formula?
A: The constants (1+2√2)/4 and √(543+176√2) are derived from the geometric properties and trigonometric relationships specific to the Hexakis Octahedron.