Surface To Volume Ratio Of Pentakis Dodecahedron Given Volume Calculator

Formula Used:

\[ \text{Surface to Volume Ratio} = \frac{\frac{76}{19} \times \sqrt{413 + 162 \times \sqrt{5}}}{23 + 11 \times \sqrt{5}} \times \left( \frac{15 \times (23 + 11 \times \sqrt{5})}{76 \times \text{Volume}} \right)^{\frac{1}{3}} \]

Surface to Volume Ratio:

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1. What is Surface to Volume Ratio of Pentakis Dodecahedron?

The Surface to Volume Ratio of a Pentakis Dodecahedron is a geometric property that represents the relationship between the total surface area and the volume of this polyhedron. It indicates how much surface area is available per unit volume of the shape.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

Where:

Volume — Volume of the Pentakis Dodecahedron (m³)
√5 — Square root of 5 (approximately 2.23607)

Explanation: The formula combines geometric constants and the cube root of the inverse volume to calculate the surface to volume ratio.

3. Importance of Surface to Volume Ratio Calculation

Details: The surface to volume ratio is important in various fields including materials science, chemistry, and physics. For geometric shapes, it helps understand how the shape's properties scale with size and is useful in optimization problems.

4. Using the Calculator

Tips: Enter the volume of the Pentakis Dodecahedron in cubic meters. The volume must be a positive value 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 that is the dual of the truncated icosahedron. It has 60 faces, 90 edges, and 32 vertices.

Q2: Why is the surface to volume ratio important?
A: This ratio is crucial in understanding how properties like heat transfer, chemical reactivity, and mechanical strength scale with the size of the object.

Q3: What units should I use for volume?
A: The calculator expects volume in cubic meters (m³). If you have volume in other units, convert it to cubic meters first.

Q4: Can this calculator handle very small or very large volumes?
A: Yes, the calculator can handle a wide range of volume values, but extremely small values close to zero may cause computational issues.

Q5: Is this formula specific to Pentakis Dodecahedron?
A: Yes, this formula is specifically derived for calculating the surface to volume ratio of a Pentakis Dodecahedron given its volume.