Midsphere Radius Of Hexakis Octahedron Given Short Edge Calculator

Formula Used:

\[ r_m = \frac{1 + 2\sqrt{2}}{4} \times \frac{14 \times l_e(Short)}{10 - \sqrt{2}} \]

Midsphere Radius of Hexakis Octahedron:

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1. What is the Midsphere Radius of Hexakis Octahedron?

The Midsphere Radius of 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 represents the sphere that touches the midpoints of all edges of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{1 + 2\sqrt{2}}{4} \times \frac{14 \times l_e(Short)}{10 - \sqrt{2}} \]

Where:

\( r_m \) — Midsphere Radius of Hexakis Octahedron (m)
\( l_e(Short) \) — Short Edge of Hexakis Octahedron (m)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)

Explanation: This formula calculates the midsphere radius based on the length of the shortest edge 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 geometric analysis and 3D modeling of polyhedrons. It helps in understanding the spatial properties and symmetry of the Hexakis Octahedron, which is useful in various mathematical and engineering applications.

4. Using the Calculator

Tips: Enter the Short Edge length in meters. The value must be positive and valid. The calculator will compute the corresponding midsphere radius using the established geometric formula.

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

Q2: Why is the midsphere radius important?
A: The midsphere radius helps in understanding the geometric properties and symmetry of the polyhedron, which is useful in various mathematical, architectural, and engineering applications.

Q3: What are typical values for the Short Edge?
A: The Short Edge length depends on the specific Hexakis Octahedron being considered. It can vary based on the scale and dimensions of the polyhedron.

Q4: Are there limitations to this formula?
A: This formula is specifically derived for the Hexakis Octahedron and assumes ideal geometric conditions. It may not apply to other polyhedrons or modified shapes.

Q5: Can this calculator be used for educational purposes?
A: Yes, this calculator is useful for students and educators studying polyhedral geometry and spatial mathematics.