Total Surface Area of Pentakis Dodecahedron given Midsphere Radius Calculator

Formula Used:

\[ TSA = \frac{15}{19} \times \sqrt{413 + (162 \times \sqrt{5})} \times \left( \frac{4 \times r_m}{3 + \sqrt{5}} \right)^2 \]

Total Surface Area of Pentakis Dodecahedron:

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1. What is the Total Surface Area of Pentakis Dodecahedron?

The Total Surface Area of Pentakis Dodecahedron is the total area of all the faces of a Pentakis Dodecahedron, which is a Catalan solid derived from the dodecahedron by placing a pyramid on each of its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \frac{15}{19} \times \sqrt{413 + (162 \times \sqrt{5})} \times \left( \frac{4 \times r_m}{3 + \sqrt{5}} \right)^2 \]

Where:

\( TSA \) — Total Surface Area of Pentakis Dodecahedron (m²)
\( r_m \) — Midsphere Radius of Pentakis Dodecahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the total surface area based on the midsphere radius of the Pentakis Dodecahedron, incorporating mathematical constants and operations.

3. Importance of Surface Area Calculation

Details: Calculating the surface area is important in geometry and various applications such as material science, architecture, and 3D modeling where surface coverage or material requirements need to be determined.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid that is the dual of the truncated icosahedron. It has 60 faces, 90 edges, and 32 vertices.

Q2: What is the midsphere radius?
A: The midsphere radius is the radius of the sphere that is tangent to all the edges of the Pentakis Dodecahedron.

Q3: Are there other ways to calculate the surface area?
A: Yes, the surface area can also be calculated using other parameters such as edge length or volume, but this calculator specifically uses the midsphere radius.

Q4: What units should I use?
A: The calculator uses meters for input and square meters for output. Ensure consistent units for accurate results.

Q5: Can this calculator handle very large or very small values?
A: The calculator is designed to handle a wide range of positive values, but extremely large or small numbers might be limited by PHP's floating-point precision.