Volume Of Pentakis Dodecahedron Given Surface To Volume Ratio Calculator

Formula Used:

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

Volume of Pentakis Dodecahedron:

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1. What is Volume of Pentakis Dodecahedron?

The volume of a Pentakis Dodecahedron is the quantity of three dimensional space enclosed by the entire surface of this polyhedron. It's a Catalan solid that can be derived from a dodecahedron by placing a pyramid on each face.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( V \) — Volume of Pentakis Dodecahedron (m³)
\( RA/V \) — Surface to Volume Ratio (1/m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the volume based on the surface to volume ratio, using mathematical constants derived from the geometry of the Pentakis Dodecahedron.

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 properties are crucial.

4. Using the Calculator

Tips: Enter the surface to volume ratio in 1/m. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid with 60 isosceles triangular faces, 90 edges, and 32 vertices. It's the dual of the truncated icosahedron.

Q2: What are typical surface to volume ratio values?
A: The surface to volume ratio depends on the size of the polyhedron. Smaller objects have higher ratios, while larger objects have lower ratios.

Q3: Can this calculator handle different units?
A: The calculator uses meters as the base unit. For other units, convert your measurements to meters first or adjust the result accordingly.

Q4: What is the precision of the calculation?
A: The calculator provides results with 6 decimal places precision, which is suitable for most practical applications.

Q5: Are there limitations to this formula?
A: This formula is specifically designed for perfect Pentakis Dodecahedrons and assumes ideal geometric properties.