1. What is the Volume of Pentakis Dodecahedron?

The Pentakis Dodecahedron is a Catalan solid derived from the dodecahedron by placing a pyramid on each face. Its volume represents the three-dimensional space enclosed by its surface, calculated based on geometric properties.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( V \) — Volume of Pentakis Dodecahedron (m³)
• \( r_m \) — Midsphere Radius (m)
• \( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: The formula calculates the volume based on the midsphere radius, incorporating mathematical constants and geometric relationships specific to the Pentakis Dodecahedron.

3. Importance of Volume Calculation

Details: Accurate volume calculation is essential for understanding the spatial properties of the Pentakis Dodecahedron, useful in fields like crystallography, architecture, and mathematical modeling.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and valid. The calculator will compute the volume using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: It is a convex polyhedron with 60 faces, derived from the dodecahedron by adding a pyramid to each face, resulting in a structure with 32 vertices and 90 edges.

Q2: How is the midsphere radius defined?
A: The midsphere radius is the radius of the sphere that is tangent to all edges of the Pentakis Dodecahedron.

Q3: What are typical values for the midsphere radius?
A: The midsphere radius depends on the specific dimensions of the Pentakis Dodecahedron, but it is always a positive real number.

Q4: Can this formula be used for other polyhedra?
A: No, this formula is specific to the Pentakis Dodecahedron due to its unique geometric properties.

Q5: Why is the square root of 5 involved?
A: The square root of 5 appears naturally in formulas related to pentagonal symmetry, which is inherent to the dodecahedron and its derivatives.