Volume of Pentakis Dodecahedron given Insphere Radius Calculator

Formula Used:

\[ V = \frac{15}{76} \times (23 + 11\sqrt{5}) \times \left(\frac{2 \times r_i}{3 \times \sqrt{\frac{81 + 35\sqrt{5}}{218}}}\right)^3 \]

Volume of Pentakis Dodecahedron:

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1. What is the Volume of Pentakis Dodecahedron given Insphere Radius?

The Volume of Pentakis Dodecahedron given Insphere Radius is the quantity of three dimensional space enclosed by the entire surface of Pentakis Dodecahedron, calculated using the insphere radius.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{15}{76} \times (23 + 11\sqrt{5}) \times \left(\frac{2 \times r_i}{3 \times \sqrt{\frac{81 + 35\sqrt{5}}{218}}}\right)^3 \]

Where:

\( V \) — Volume of Pentakis Dodecahedron (m³)
\( r_i \) — Insphere Radius of Pentakis Dodecahedron (m)
\( \sqrt{} \) — Square root function

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

3. Importance of Volume Calculation

Details: Accurate volume calculation is crucial for understanding the spatial properties of the Pentakis Dodecahedron, which has applications in geometry, architecture, and material science.

4. Using the Calculator

Tips: Enter the insphere radius 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 is the dual of the truncated icosahedron, featuring 60 isosceles triangular faces.

Q2: How is the insphere radius defined?
A: The insphere radius is the radius of the sphere that is contained by the Pentakis Dodecahedron such that all faces just touch the sphere.

Q3: What are the units of measurement?
A: The input (insphere radius) should be in meters, and the output (volume) will be in cubic meters (m³).

Q4: Are there limitations to this formula?
A: The formula is mathematically derived and accurate for all valid positive values of the insphere radius.

Q5: Can this calculator be used for educational purposes?
A: Yes, this calculator is designed to help students and professionals quickly compute the volume of a Pentakis Dodecahedron given its insphere radius.