Midsphere Radius of Truncated Dodecahedron Given Dodecahedral Edge Length Calculator

Formula Used:

\[ r_m = \frac{5 + 3\sqrt{5}}{4} \times \frac{l_{e(dodecahedron)}}{\sqrt{5}} \]

Midsphere Radius:

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1. What is the Midsphere Radius of Truncated Dodecahedron?

The Midsphere Radius of a Truncated Dodecahedron is the radius of the sphere that is tangent to all the edges of the Truncated Dodecahedron. It represents the sphere that fits perfectly within the polyhedron, touching each edge at exactly one point.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_m = \frac{5 + 3\sqrt{5}}{4} \times \frac{l_{e(dodecahedron)}}{\sqrt{5}} \]

Where:

\( r_m \) — Midsphere Radius (m)
\( l_{e(dodecahedron)} \) — Dodecahedral Edge Length (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the midsphere radius based on the dodecahedral edge length of the truncated dodecahedron, incorporating the mathematical constant √5.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and 3D modeling for understanding the spatial properties of truncated dodecahedrons and their relationship with inscribed spheres.

4. Using the Calculator

Tips: Enter the dodecahedral edge length in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Truncated Dodecahedron?
A: A Truncated Dodecahedron is an Archimedean solid created by cutting the corners of a regular dodecahedron, resulting in a polyhedron with 20 regular triangular faces and 12 regular decagonal faces.

Q2: How is the Midsphere Radius different from the Insphere Radius?
A: The Midsphere Radius is tangent to all edges, while the Insphere Radius is tangent to all faces of the polyhedron.

Q3: What are practical applications of this calculation?
A: This calculation is used in geometry research, 3D modeling, architectural design, and in understanding the properties of polyhedral structures.

Q4: Can this formula be used for other polyhedrons?
A: No, this specific formula applies only to Truncated Dodecahedrons. Other polyhedrons have different formulas for calculating their midsphere radii.

Q5: What is the significance of the √5 constant in the formula?
A: The √5 constant appears frequently in formulas related to dodecahedrons and icosahedrons due to their relationship with the golden ratio φ = (1+√5)/2.