Insphere Radius of Pentakis Dodecahedron given Midsphere Radius Calculator

Formula Used:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{81 + 35\sqrt{5}}{218}} \times \frac{4r_m}{3 + \sqrt{5}} \]

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 being tangent to all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{3}{2} \times \sqrt{\frac{81 + 35\sqrt{5}}{218}} \times \frac{4r_m}{3 + \sqrt{5}} \]

Where:

\( r_i \) — Insphere Radius of Pentakis Dodecahedron (m)
\( r_m \) — Midsphere Radius of Pentakis Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the insphere radius based on the given midsphere radius, using the mathematical relationship between these two geometric properties of the Pentakis Dodecahedron.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry and 3D modeling as it helps determine the maximum size of a sphere that can be inscribed within the Pentakis Dodecahedron without intersecting any of its faces.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentakis Dodecahedron?
A: A Pentakis Dodecahedron is a Catalan solid that is the dual of the truncated icosahedron. It has 60 isosceles triangular faces.

Q2: What is the difference between insphere radius and midsphere radius?
A: The insphere radius is the radius of the sphere inscribed within the polyhedron (touching all faces), while the midsphere radius is the radius of the sphere that touches all edges of the polyhedron.

Q3: What are the units of measurement?
A: Both radii are measured in meters (m), though any consistent unit of length can be used as long as the same unit is maintained throughout the calculation.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to the Pentakis Dodecahedron. Other polyhedra have different mathematical relationships between their insphere and midsphere radii.

Q5: What is the accuracy of the calculation?
A: The calculation provides results accurate to 10 decimal places, which is sufficient for most geometric and engineering applications.