Volume Of Pentakis Dodecahedron Given Leg Length Calculator

Formula Used:

\[ V = \frac{15}{76} \times (23 + 11\sqrt{5}) \times \left( \frac{38 \times l_{Leg}}{3 \times (9 + \sqrt{5})} \right)^3 \]

Volume of Pentakis Dodecahedron:

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1. What is the Volume of Pentakis Dodecahedron?

The Pentakis Dodecahedron is a Catalan solid that is the dual of the truncated icosahedron. Its volume represents the three-dimensional space enclosed by all its faces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{15}{76} \times (23 + 11\sqrt{5}) \times \left( \frac{38 \times l_{Leg}}{3 \times (9 + \sqrt{5})} \right)^3 \]

Where:

\( V \) — Volume of Pentakis Dodecahedron (m³)
\( l_{Leg} \) — Leg Length of Pentakis Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the volume based on the leg length of the isosceles triangular faces of the Pentakis Dodecahedron.

3. Importance of Volume Calculation

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

4. Using the Calculator

Tips: Enter the leg 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 with 60 isosceles triangular faces, 90 edges, and 32 vertices.

Q2: How is this different from a regular dodecahedron?
A: While a regular dodecahedron has 12 regular pentagonal faces, the Pentakis Dodecahedron has each pentagonal face replaced by five isosceles triangles.

Q3: What are the practical applications of this calculation?
A: This calculation is used in geometric modeling, 3D printing, architectural design, and in understanding molecular structures in chemistry.

Q4: Can this formula be used for any Pentakis Dodecahedron?
A: Yes, this formula applies to all Pentakis Dodecahedrons as long as the leg length is known.

Q5: How accurate is the calculated volume?
A: The calculation is mathematically exact based on the given formula. The accuracy depends on the precision of the input leg length measurement.