Insphere Radius of Deltoidal Hexecontahedron given NonSymmetry Diagonal Calculator

Formula:

\[ r_i = \frac{3}{2} \cdot \sqrt{\frac{135 + 59\sqrt{5}}{205}} \cdot \frac{11 \cdot d_{NonSymmetry}}{\sqrt{\frac{470 + 156\sqrt{5}}{5}}} \]

Insphere Radius of Deltoidal Hexecontahedron:

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1. What is Insphere Radius of Deltoidal Hexecontahedron?

The Insphere Radius of a Deltoidal Hexecontahedron is the radius of the sphere that is contained by the Deltoidal Hexecontahedron in such a way that all the faces just touch the sphere. This geometric property is important in understanding the spatial characteristics of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{3}{2} \cdot \sqrt{\frac{135 + 59\sqrt{5}}{205}} \cdot \frac{11 \cdot d_{NonSymmetry}}{\sqrt{\frac{470 + 156\sqrt{5}}{5}}} \]

Where:

\( r_i \) — Insphere Radius of Deltoidal Hexecontahedron (m)
\( d_{NonSymmetry} \) — NonSymmetry Diagonal of Deltoidal Hexecontahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the insphere radius based on the non-symmetry diagonal measurement, incorporating mathematical constants specific to the geometry of the deltoidal hexecontahedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is crucial for understanding the internal geometry of the deltoidal hexecontahedron, which has applications in crystallography, molecular modeling, and architectural design where this specific polyhedral form is utilized.

4. Using the Calculator

Tips: Enter the NonSymmetry Diagonal measurement in meters. The value must be positive and greater than zero. The calculator will compute the corresponding insphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Deltoidal Hexecontahedron?
A: A Deltoidal Hexecontahedron is a Catalan solid with 60 deltoid (kite-shaped) faces. It is the dual polyhedron of the rhombicosidodecahedron.

Q2: What is the NonSymmetry Diagonal?
A: The NonSymmetry Diagonal of a Deltoidal Hexecontahedron is the length of the diagonal which divides the deltoid faces into two isosceles triangles.

Q3: How accurate is this calculation?
A: The calculation is mathematically precise based on the geometric properties of the deltoidal hexecontahedron, assuming exact input values.

Q4: What are typical values for the NonSymmetry Diagonal?
A: The NonSymmetry Diagonal varies depending on the size of the polyhedron. There's no "typical" value as it depends on the specific dimensions of the shape being studied.

Q5: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the deltoidal hexecontahedron. Other polyhedra have different formulas for calculating their insphere radii.