Total Surface Area of Pentagonal Hexecontahedron given Midsphere Radius Calculator

Formula Used:

\[ TSA = 30 \times \left( \frac{r_m}{\sqrt{\frac{1+0.4715756}{2 \times (1-2 \times 0.4715756)}}} \right)^2 \times (2+3 \times 0.4715756) \times \frac{\sqrt{1-0.4715756^2}}{1-2 \times 0.4715756^2} \]

Total Surface Area of Pentagonal Hexecontahedron (TSA):

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1. What is the Total Surface Area of Pentagonal Hexecontahedron?

The Total Surface Area of Pentagonal Hexecontahedron is the amount or quantity of two dimensional space covered on the surface of Pentagonal Hexecontahedron. It is an important geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ TSA = 30 \times \left( \frac{r_m}{\sqrt{\frac{1+0.4715756}{2 \times (1-2 \times 0.4715756)}}} \right)^2 \times (2+3 \times 0.4715756) \times \frac{\sqrt{1-0.4715756^2}}{1-2 \times 0.4715756^2} \]

Where:

\( TSA \) — Total Surface Area of Pentagonal Hexecontahedron (m²)
\( r_m \) — Midsphere Radius of Pentagonal Hexecontahedron (m)
0.4715756 — Constant value used in the calculation

Explanation: This formula calculates the total surface area based on the midsphere radius using specific mathematical constants and operations.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of geometric shapes is crucial in various fields including architecture, engineering, material science, and mathematical research. It helps in determining material requirements, heat transfer properties, and other physical characteristics.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and valid. The calculator will compute the total surface area using the specialized formula for pentagonal hexecontahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Hexecontahedron?
A: A Pentagonal Hexecontahedron is a complex polyhedron with 60 pentagonal faces. It is one of the Catalan solids and is the dual of the snub dodecahedron.

Q2: What is the Midsphere Radius?
A: The Midsphere Radius is the radius of the sphere that is tangent to all edges of the polyhedron. It is an important geometric property used in various calculations.

Q3: Why are there specific constants in the formula?
A: The constants (like 0.4715756) are derived from the specific geometric properties and mathematical relationships of the pentagonal hexecontahedron shape.

Q4: What units should I use for input?
A: The calculator expects the midsphere radius in meters (m), and returns the surface area in square meters (m²). Ensure consistent units for accurate results.

Q5: Can this calculator be used for other polyhedra?
A: No, this specific calculator is designed only for pentagonal hexecontahedron. Other polyhedra have different formulas for surface area calculation.