1. What is Circumsphere Radius of Dodecahedron?

The Circumsphere Radius of a Dodecahedron is the radius of the sphere that contains the Dodecahedron in such a way that all the vertices are lying on the sphere. It is an important geometric property of this regular polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( r_c \) — Circumsphere Radius of Dodecahedron (m)
• \( r_i \) — Insphere Radius of Dodecahedron (m)

3. Formula Explanation

Details: This formula establishes the mathematical relationship between the circumsphere radius and insphere radius of a regular dodecahedron, using the golden ratio (φ = (1+√5)/2) and other geometric constants specific to this polyhedron.

4. Using the Calculator

Tips: Enter the insphere radius value in meters. The value must be positive and greater than zero. The calculator will compute the corresponding circumsphere radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a Dodecahedron?
A: A dodecahedron is a regular polyhedron with 12 identical pentagonal faces, 20 vertices, and 30 edges.

Q2: What is the difference between circumsphere and insphere?
A: The circumsphere passes through all vertices, while the insphere is tangent to all faces of the polyhedron.

Q3: Can this formula be used for irregular dodecahedrons?
A: No, this formula applies only to regular dodecahedrons where all faces are identical regular pentagons.

Q4: What are typical values for these radii?
A: For a unit dodecahedron (edge length = 1), the circumsphere radius is approximately 1.401258 while the insphere radius is approximately 1.113516.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for perfect regular dodecahedrons, with precision limited only by computational floating-point arithmetic.