1. What is the Perimeter of Dodecahedron given Circumsphere Radius Formula?

The formula calculates the perimeter of a dodecahedron based on its circumsphere radius. A dodecahedron is a three-dimensional shape with 12 regular pentagonal faces, 20 vertices, and 30 edges.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( P \) — Perimeter of Dodecahedron (m)
• \( r_c \) — Circumsphere Radius of Dodecahedron (m)
• \( \sqrt{3} \) — Square root of 3 (approximately 1.732)
• \( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: The formula derives from the geometric properties of a regular dodecahedron and its relationship with its circumscribed sphere.

3. Importance of Perimeter Calculation

Details: Calculating the perimeter of a dodecahedron is important in geometry, 3D modeling, and various engineering applications where precise measurements of polyhedral structures are required.

4. Using the Calculator

Tips: Enter the circumsphere radius in meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

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

Q2: What is the circumsphere radius?
A: The circumsphere radius is the radius of the sphere that passes through all vertices of the dodecahedron.

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 practical applications of this calculation?
A: This calculation is used in architecture, 3D modeling, crystallography, and various engineering fields.

Q5: How accurate is this formula?
A: The formula is mathematically exact for regular dodecahedrons and provides precise results when accurate inputs are provided.