Midsphere Radius Of Hexakis Octahedron Calculator

Midsphere Radius of Hexakis Octahedron Formula:

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

Midsphere Radius of Hexakis Octahedron (r_m):

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1. What is 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 all the edges of the polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_m \) — Midsphere Radius of Hexakis Octahedron (meters)
\( l_e \) — Long Edge of Hexakis Octahedron (meters)
\( \sqrt{2} \) — Square root of 2 (approximately 1.414213562)

Explanation: The formula calculates the radius of the midsphere based on the long edge length of the Hexakis Octahedron, incorporating the mathematical constant √2.

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 determining the sphere that would perfectly fit around the polyhedron, touching all its edges.

4. Using the Calculator

Tips: Enter the Long Edge of Hexakis Octahedron in meters. The value must be positive and greater than zero. The calculator will compute the corresponding Midsphere Radius.

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: How is the midsphere different from the insphere?
A: The midsphere touches all the edges of the polyhedron, while the insphere touches all the faces from inside.

Q3: What are typical values for the long edge?
A: The long edge length depends on the specific Hexakis Octahedron being measured. It can vary from millimeters to meters depending on the application.

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

Q5: What is the significance of √2 in the formula?
A: √2 appears in the formula due to the geometric properties and symmetry of the Hexakis Octahedron, particularly related to its relationship with the cube and octahedron.