Midsphere Radius of Pentagonal Icositetrahedron Given Surface to Volume Ratio Calculator

Formula:

\[ r_m = \frac{3\sqrt{\frac{22(5[Tribonacci_C]-1)}{(4[Tribonacci_C])-3}}}{2 \times \frac{SA}{V} \times \sqrt{\frac{11([Tribonacci_C]-4)}{2((20[Tribonacci_C])-37)}} \times \sqrt{2-[Tribonacci_C]}}} \]

Midsphere Radius (rₘ):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Midsphere Radius of Pentagonal Icositetrahedron?

The midsphere radius of a Pentagonal Icositetrahedron 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 while touching all its edges.

2. How Does the Calculator Work?

The calculator uses the specialized formula:

\[ r_m = \frac{3\sqrt{\frac{22(5[Tribonacci_C]-1)}{(4[Tribonacci_C])-3}}}{2 \times \frac{SA}{V} \times \sqrt{\frac{11([Tribonacci_C]-4)}{2((20[Tribonacci_C])-37)}} \times \sqrt{2-[Tribonacci_C]}}} \]

Where:

\( r_m \) — Midsphere radius (m)
\( \frac{SA}{V} \) — Surface to volume ratio (1/m)
\( [Tribonacci_C] \) — Tribonacci constant ≈ 1.839286755214161

Explanation: This formula relates the midsphere radius to the surface-to-volume ratio using the mathematical constant specific to pentagonal icositetrahedron geometry.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometric modeling, crystallography, and understanding the spatial properties of this specific polyhedral form. It helps in determining optimal packing and spatial relationships.

4. Using the Calculator

Tips: Enter the surface-to-volume ratio in 1/m. The value must be positive and valid for the specific geometry of a pentagonal icositetrahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Icositetrahedron?
A: A Pentagonal Icositetrahedron is a Catalan solid with 24 pentagonal faces, 60 edges, and 38 vertices. It is the dual of the snub cube.

Q2: What is the Tribonacci constant?
A: The Tribonacci constant is the ratio toward which adjacent Tribonacci numbers tend. It is the real root of the equation x³ = x² + x + 1.

Q3: How is the surface-to-volume ratio measured?
A: The surface-to-volume ratio is calculated as the total surface area divided by the total volume of the polyhedron, typically expressed in 1/m units.

Q4: What are typical values for this calculation?
A: Values depend on the specific dimensions of the pentagonal icositetrahedron, but the surface-to-volume ratio typically ranges based on the scale of the polyhedron.

Q5: Can this formula be applied to other polyhedra?
A: No, this specific formula is derived for the pentagonal icositetrahedron geometry and uses the Tribonacci constant which is unique to this form.