Volume of Pentagonal Icositetrahedron given Midsphere Radius Calculator

Formula Used:

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

Volume of Pentagonal Icositetrahedron (V):

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1. What is the Volume of Pentagonal Icositetrahedron?

The volume of a Pentagonal Icositetrahedron is the quantity of three-dimensional space enclosed by the entire surface of this polyhedron. It is a Catalan solid with 24 pentagonal faces.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( V \) — Volume of Pentagonal Icositetrahedron (m³)
\( r_m \) — Midsphere Radius (m)
\( [Tribonacci_C] \) — Tribonacci constant (≈1.839286755214161)

Explanation: This formula relates the volume of the Pentagonal Icositetrahedron to its midsphere radius using the mathematical constant Tribonacci_C.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, physics, engineering, and various scientific applications where spatial measurements and properties are required.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and valid. The calculator will compute the volume using the established mathematical formula.

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 real root of the equation x³ - x² - x - 1 = 0, approximately equal to 1.839286755214161.

Q3: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the polyhedron.

Q4: What are the applications of this calculation?
A: This calculation is used in geometry, crystallography, architectural design, and mathematical research involving polyhedral structures.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the given formula and the defined Tribonacci constant value.