Volume Of Pentagonal Hexecontahedron Given Insphere Radius Calculator

Volume of Pentagonal Hexecontahedron Formula:

\[ V = 5 \times \left( \frac{r_i \times 2}{\sqrt{\frac{1+0.4715756}{(1-0.4715756) \times (1-2 \times 0.4715756)}}} \right)^3 \times \frac{(1+0.4715756) \times (2+3 \times 0.4715756)}{(1-2 \times 0.4715756^2) \times \sqrt{1-2 \times 0.4715756}} \]

Volume of Pentagonal Hexecontahedron:

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1. What is Volume of Pentagonal Hexecontahedron?

The Pentagonal Hexecontahedron is a Catalan solid with 60 pentagonal faces. Its volume represents the three-dimensional space enclosed by all its faces, calculated based on the insphere radius.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( V \) — Volume of Pentagonal Hexecontahedron (m³)
\( r_i \) — Insphere Radius of Pentagonal Hexecontahedron (m)
0.4715756 — Constant value specific to the geometry

Explanation: The formula calculates the volume based on the insphere radius using geometric relationships and mathematical constants specific to the pentagonal hexecontahedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is essential in various fields including mathematics, engineering, architecture, and material science for understanding spatial properties and capacity.

4. Using the Calculator

Tips: Enter the insphere radius in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Hexecontahedron?
A: A Pentagonal Hexecontahedron is a Catalan solid with 60 pentagonal faces, 150 edges, and 92 vertices.

Q2: What is the insphere radius?
A: The insphere radius is the radius of the largest sphere that can fit inside the polyhedron, touching all faces.

Q3: Why is the constant 0.4715756 used?
A: This constant is derived from the geometric properties and mathematical relationships specific to the pentagonal hexecontahedron.

Q4: What are the units of measurement?
A: The calculator uses meters for input (insphere radius) and cubic meters for output (volume).

Q5: Can this calculator handle very large or small values?
A: Yes, but extremely large or small values may be limited by floating-point precision in computational systems.