Midsphere Radius Of Pentagonal Icositetrahedron Given Total Surface Area Calculator

Formula Used:

\[ r_m = \frac{1}{2\sqrt{2 - C_{Tribonacci}}}} \times \sqrt{\frac{TSA}{3}} \times \left( \frac{(4 \times C_{Tribonacci}) - 3}{22 \times ((5 \times C_{Tribonacci}) - 1)} \right)^{\frac{1}{4}} \]

Midsphere Radius (rₘ):

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1. What is Midsphere Radius of Pentagonal Icositetrahedron?

The Midsphere Radius of a Pentagonal Icositetrahedron is the radius of the sphere that is tangent to all the 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 following formula:

\[ r_m = \frac{1}{2\sqrt{2 - C_{Tribonacci}}}} \times \sqrt{\frac{TSA}{3}} \times \left( \frac{(4 \times C_{Tribonacci}) - 3}{22 \times ((5 \times C_{Tribonacci}) - 1)} \right)^{\frac{1}{4}} \]

Where:

\( r_m \) — Midsphere Radius (m)
\( TSA \) — Total Surface Area (m²)
\( C_{Tribonacci} \) — Tribonacci constant ≈ 1.839286755214161

Explanation: This formula relates the midsphere radius to the total surface area of the pentagonal icositetrahedron using the mathematical constant Tribonacci constant.

3. Importance of Midsphere Radius Calculation

Details: Calculating the midsphere radius is important in geometry and crystallography for understanding the spatial properties and symmetry of complex polyhedra like the pentagonal icositetrahedron.

4. Using the Calculator

Tips: Enter the total surface area of the pentagonal icositetrahedron in square meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Icositetrahedron?
A: A pentagonal icositetrahedron is a Catalan solid with 24 pentagonal faces, 38 vertices, and 60 edges. 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: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the pentagonal icositetrahedron and the defined formula.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the pentagonal icositetrahedron due to its unique geometric properties.

Q5: What are practical applications of this calculation?
A: This calculation is used in mathematical research, crystallography, and the study of geometric properties of complex polyhedra.