Insphere Radius of Pentakis Dodecahedron Calculator

Formula Used:

\[ r_i = \frac{3 \cdot \sqrt{\frac{81 + 35 \cdot \sqrt{5}}{218}} \cdot l_{Base}}{2} \]

Insphere Radius of Pentakis Dodecahedron:

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1. What is the Insphere Radius of Pentakis Dodecahedron?

The Insphere Radius of Pentakis Dodecahedron is the radius of the sphere that is contained by the Pentakis Dodecahedron in such a way that all the faces just touch the sphere. It represents the largest sphere that can fit inside the polyhedron while touching all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{3 \cdot \sqrt{\frac{81 + 35 \cdot \sqrt{5}}{218}} \cdot l_{Base}}{2} \]

Where:

\( r_i \) — Insphere Radius of Pentakis Dodecahedron (m)
\( l_{Base} \) — Base Length of Pentakis Dodecahedron (m)
\( \sqrt{} \) — Square root function

Explanation: This formula calculates the radius of the inscribed sphere based on the base length of the Pentakis Dodecahedron, incorporating mathematical constants related to the golden ratio and the geometry of the shape.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and material science for understanding the packing efficiency, structural properties, and spatial relationships within the Pentakis Dodecahedron shape.

4. Using the Calculator

Tips: Enter the base length of the Pentakis Dodecahedron in meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid that can be seen as a dodecahedron with a pyramid on each face, creating a polyhedron with 60 faces.

Q2: How is this formula derived?
A: The formula is derived from the geometric properties and trigonometric relationships within the Pentakis Dodecahedron structure.

Q3: What are practical applications of this calculation?
A: This calculation is used in crystallography, architectural design, and the study of geometric properties in various scientific fields.

Q4: Can this calculator handle different units?
A: The calculator uses meters as the default unit. For other units, convert your measurement to meters before calculation.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the given formula, with precision limited only by computational floating-point arithmetic.