Short Edge of Pentagonal Hexecontahedron Given Midsphere Radius Calculator

Formula Used:

\[ \text{Short Edge} = \frac{r_m}{\sqrt{\frac{1 + 0.4715756}{2 \times (1 - 2 \times 0.4715756)}}} \]

Short Edge of Pentagonal Hexecontahedron:

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1. What is the Short Edge of Pentagonal Hexecontahedron?

The Short Edge of Pentagonal Hexecontahedron is the length of the shortest edge which is the base and middle edge of the axial-symmetric pentagonal faces of the Pentagonal Hexecontahedron. It is an important geometric property of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Short Edge} = \frac{r_m}{\sqrt{\frac{1 + 0.4715756}{2 \times (1 - 2 \times 0.4715756)}}} \]

Where:

\( r_m \) — Midsphere Radius of Pentagonal Hexecontahedron (m)
0.4715756 — A constant specific to the geometry of Pentagonal Hexecontahedron

Explanation: This formula calculates the short edge length based on the midsphere radius, using the specific geometric properties of the Pentagonal Hexecontahedron.

3. Importance of Short Edge Calculation

Details: Calculating the short edge is essential for understanding the complete geometry of the Pentagonal Hexecontahedron, its surface area, volume, and other geometric properties. It's particularly important in crystallography and materials science applications.

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 short edge length of the Pentagonal Hexecontahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Hexecontahedron?
A: A Pentagonal Hexecontahedron is a Catalan solid with 60 irregular pentagonal faces. It is the dual polyhedron of the snub dodecahedron.

Q2: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q3: Why is the constant 0.4715756 used?
A: This constant is derived from the specific geometric properties and trigonometric relationships within the Pentagonal Hexecontahedron structure.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Pentagonal Hexecontahedron due to its unique geometric properties.

Q5: What are typical values for the short edge?
A: The short edge length varies depending on the size of the polyhedron, but it's typically smaller than other edges in the Pentagonal Hexecontahedron structure.