Midsphere Radius Of Deltoidal Hexecontahedron Given Total Surface Area Calculator

Midsphere Radius of Deltoidal Hexecontahedron Formula:

\[ r_m = \frac{3}{20} \times (5 + 3\sqrt{5}) \times \sqrt{\frac{11 \times TSA}{9 \times \sqrt{10 \times (157 + 31\sqrt{5})}}} \]

Midsphere Radius (rₘ):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Midsphere Radius of Deltoidal Hexecontahedron?

The Midsphere Radius of a Deltoidal Hexecontahedron is the radius of the sphere that is tangent to all edges of the polyhedron. 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 mathematical formula:

\[ r_m = \frac{3}{20} \times (5 + 3\sqrt{5}) \times \sqrt{\frac{11 \times TSA}{9 \times \sqrt{10 \times (157 + 31\sqrt{5})}}} \]

Where:

\( r_m \) — Midsphere Radius (m)
\( TSA \) — Total Surface Area of Deltoidal Hexecontahedron (m²)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the midsphere radius based on the total surface area of the deltoidal hexecontahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

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 deltoidal hexecontahedron, determining optimal packing arrangements, and analyzing the polyhedron's symmetry properties.

4. Using the Calculator

Tips: Enter the total surface area of the deltoidal hexecontahedron in square meters. The value must be positive and greater than zero for accurate calculation.

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, 62 vertices, and 120 edges.

Q2: How is the midsphere radius different from the insphere radius?
A: The midsphere radius touches all edges, while the insphere radius touches all faces. They represent different spheres within the polyhedron.

Q3: What are typical values for midsphere radius?
A: The midsphere radius depends on the size of the polyhedron. For a deltoidal hexecontahedron with unit surface area, the midsphere radius is approximately 0.17 units.

Q4: 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 midsphere radii.

Q5: What practical applications does this calculation have?
A: This calculation is used in crystallography, architectural design, computer graphics, and mathematical research involving polyhedral geometry.