Volume Of Pentakis Dodecahedron Given Total Surface Area Calculator

Formula Used:

\[ V = \frac{15}{76} \times (23 + 11\sqrt{5}) \times \left( \frac{19 \times TSA}{15 \times \sqrt{413 + 162\sqrt{5}}} \right)^{\frac{3}{2}} \]

Volume of Pentakis Dodecahedron:

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1. What is the Volume of Pentakis Dodecahedron?

The Pentakis Dodecahedron is a Catalan solid derived from the dodecahedron by placing a pyramid on each face. Its volume represents the three-dimensional space enclosed by its surface, calculated based on its total surface area.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{15}{76} \times (23 + 11\sqrt{5}) \times \left( \frac{19 \times TSA}{15 \times \sqrt{413 + 162\sqrt{5}}} \right)^{\frac{3}{2}} \]

Where:

\( V \) — Volume of Pentakis Dodecahedron (m³)
\( TSA \) — Total Surface Area (m²)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula relates the volume of a Pentakis Dodecahedron to its total surface area through a precise mathematical relationship involving the golden ratio constant.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, architecture, and various scientific fields where spatial measurements and capacity calculations are required.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and valid. The calculator will compute the corresponding volume of the Pentakis Dodecahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid with 60 identical isosceles triangular faces, 90 edges, and 32 vertices.

Q2: How accurate is this calculation?
A: The calculation is mathematically exact when using the precise formula, though computational precision may introduce minor rounding errors.

Q3: Can this formula be used for any Pentakis Dodecahedron?
A: Yes, this formula applies to all regular Pentakis Dodecahedrons where the pyramid height creates the characteristic shape.

Q4: What units should I use?
A: Use consistent units - typically square meters for surface area and cubic meters for volume, though any consistent unit system will work.

Q5: Are there practical applications of this calculation?
A: Yes, in crystallography, architecture, 3D modeling, and any field dealing with complex polyhedral structures.