Volume of Pentagonal Hexecontahedron given Midsphere Radius Calculator

Formula Used:

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

Volume of Pentagonal Hexecontahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Volume of Pentagonal Hexecontahedron?

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

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( V \) — Volume of Pentagonal Hexecontahedron (m³)
\( r_m \) — Midsphere Radius of Pentagonal Hexecontahedron (m)
0.4715756 — Constant value specific to the Pentagonal Hexecontahedron

Explanation: The formula derives from the geometric properties of the Pentagonal Hexecontahedron and relates its volume to its midsphere radius through a specific mathematical relationship.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, and material science for understanding spatial properties, density calculations, and structural analysis.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a Pentagonal Hexecontahedron?
A: A Pentagonal Hexecontahedron is a Catalan solid with 60 faces, each being an irregular pentagon. It is the dual of the snub dodecahedron.

Q2: What is the midsphere radius?
A: The midsphere radius is the radius of a sphere that is tangent to all edges of the polyhedron.

Q3: Why is there a constant value of 0.4715756 in the formula?
A: This constant is derived from the specific geometric properties and angles of the Pentagonal Hexecontahedron and is fundamental to its mathematical description.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula is unique to the Pentagonal Hexecontahedron due to its particular geometric characteristics.

Q5: What are practical applications of this calculation?
A: This calculation is used in mathematical modeling, architectural design, and in fields studying crystal structures and molecular geometry.