Home Back

Total Surface Area of Snub Dodecahedron given Volume Calculator

Formula Used:

\[ TSA = \left(20\sqrt{3} + 3\sqrt{25 + 10\sqrt{5}}\right) \times \left(\frac{V \times 6 \times \left(3 - \left(\left(\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2}\right)^{\frac{1}{3}} + \left(\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2}\right)^{\frac{1}{3}}\right)^2\right)^{\frac{3}{2}}}{\left(12(3\phi + 1)\left(\left(\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2}\right)^{\frac{1}{3}} + \left(\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2}\right)^{\frac{1}{3}}\right)^2 - ((36\phi + 7)\left(\left(\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2}\right)^{\frac{1}{3}} + \left(\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2}\right)^{\frac{1}{3}}\right)) - (53\phi + 6)}\right)^{\frac{2}{3}} \]

m³

Unit Converter ▲

Unit Converter ▼

From: To:

1. What is the Snub Dodecahedron?

The Snub Dodecahedron is an Archimedean solid with 92 faces (12 pentagons and 80 triangles), 150 edges, and 60 vertices. It is known for its chiral properties and complex geometric structure.

2. How Does the Calculator Work?

The calculator uses the complex mathematical formula:

\[ TSA = \left(20\sqrt{3} + 3\sqrt{25 + 10\sqrt{5}}\right) \times \left(\frac{V \times 6 \times \left(3 - \left(\left(\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2}\right)^{\frac{1}{3}} + \left(\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2}\right)^{\frac{1}{3}}\right)^2\right)^{\frac{3}{2}}}{\left(12(3\phi + 1)\left(\left(\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2}\right)^{\frac{1}{3}} + \left(\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2}\right)^{\frac{1}{3}}\right)^2 - ((36\phi + 7)\left(\left(\frac{\phi}{2} + \frac{\sqrt{\phi - \frac{5}{27}}}{2}\right)^{\frac{1}{3}} + \left(\frac{\phi}{2} - \frac{\sqrt{\phi - \frac{5}{27}}}{2}\right)^{\frac{1}{3}}\right)) - (53\phi + 6)}\right)^{\frac{2}{3}} \]

Where:

Explanation: This complex formula derives the surface area from the volume using the golden ratio and cubic roots, accounting for the unique geometric properties of the snub dodecahedron.

3. Importance of Surface Area Calculation

Details: Calculating the surface area of complex polyhedra like the snub dodecahedron is crucial for materials science, architectural design, and understanding the geometric properties of three-dimensional shapes.

4. Using the Calculator

Tips: Enter the volume of the snub dodecahedron in cubic meters. The value must be positive and valid. The calculator will compute the corresponding total surface area.

5. Frequently Asked Questions (FAQ)

Q1: What makes the snub dodecahedron special?
A: It's one of the chiral Archimedean solids, meaning it exists in two mirror-image forms that cannot be superimposed on each other.

Q2: Why is the golden ratio used in this formula?
A: The golden ratio appears naturally in the geometry of regular pentagons, which are fundamental components of the dodecahedron family.

Q3: What are practical applications of this calculation?
A: Used in crystallography, molecular modeling, architectural design, and mathematical research on polyhedral geometry.

Q4: How accurate is this formula?
A: The formula is mathematically exact for ideal snub dodecahedrons, though computational precision depends on the implementation.

Q5: Can this formula be simplified?
A: Due to the complex nature of the snub dodecahedron's geometry, this is the standard formula used in mathematical literature.

Total Surface Area of Snub Dodecahedron given Volume Calculator© - All Rights Reserved 2025