Insphere Radius of Dodecahedron given Volume Calculator

Formula Used:

\[ r_i = \frac{\sqrt{\frac{25+11\sqrt{5}}{10}}}{2} \times \left( \frac{4V}{15+7\sqrt{5}} \right)^{\frac{1}{3}} \]

Insphere Radius of Dodecahedron:

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1. What is the Insphere Radius of Dodecahedron?

The Insphere Radius of a Dodecahedron is the radius of the largest sphere that can be contained within the dodecahedron such that it touches all the faces of the polyhedron. It represents the distance from the center of the dodecahedron to the center of any face.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{\frac{25+11\sqrt{5}}{10}}}{2} \times \left( \frac{4V}{15+7\sqrt{5}} \right)^{\frac{1}{3}} \]

Where:

\( r_i \) — Insphere Radius of Dodecahedron (m)
\( V \) — Volume of Dodecahedron (m³)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the insphere radius based on the volume of the dodecahedron, using the mathematical constants derived from the geometric properties of this regular polyhedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry, material science, and engineering applications where understanding the internal spatial relationships of dodecahedral structures is required.

4. Using the Calculator

Tips: Enter the volume of the dodecahedron in cubic meters. The volume must be a positive value greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a dodecahedron?
A: A dodecahedron is a regular polyhedron with twelve identical regular pentagonal faces, twenty vertices, and thirty edges.

Q2: How is the insphere radius different from the circumsphere radius?
A: The insphere radius touches all faces from inside, while the circumsphere radius passes through all vertices of the dodecahedron.

Q3: What are the practical applications of this calculation?
A: This calculation is used in crystallography, architecture, game development, and any field dealing with dodecahedral structures.

Q4: Can this formula be used for irregular dodecahedrons?
A: No, this formula is specifically for regular dodecahedrons where all faces are identical regular pentagons.

Q5: What is the relationship between volume and insphere radius?
A: The insphere radius increases with the cube root of the volume, maintaining the geometric proportions of the dodecahedron.