Volume Of Truncated Dodecahedron Given Dodecahedral Edge Length Calculator

Formula Used:

\[ V = \frac{99 + 47\sqrt{5}}{12\sqrt{5}} \times l_{e(dodecahedron)}^3 \]

Volume of Truncated Dodecahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Volume of Truncated Dodecahedron?

The volume of a truncated dodecahedron is the total quantity of three dimensional space enclosed by the surface of this Archimedean solid. It is calculated based on the dodecahedral edge length from which the corners are cut to form the truncated dodecahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = \frac{99 + 47\sqrt{5}}{12\sqrt{5}} \times l_{e(dodecahedron)}^3 \]

Where:

\( V \) — Volume of the truncated dodecahedron (m³)
\( l_{e(dodecahedron)} \) — Dodecahedral edge length (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula accounts for the geometric properties of the truncated dodecahedron, which is created by cutting the corners of a regular dodecahedron.

3. Importance of Volume Calculation

Details: Calculating the volume of geometric solids is fundamental in mathematics, engineering, architecture, and various scientific fields. It helps in material estimation, structural analysis, and understanding spatial relationships.

4. Using the Calculator

Tips: Enter the dodecahedral edge length in meters. The value must be positive and greater than zero. The calculator will compute the volume using the precise mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated dodecahedron?
A: A truncated dodecahedron is an Archimedean solid created by cutting the corners of a regular dodecahedron. It has 20 regular triangular faces and 12 regular decagonal faces.

Q2: How accurate is this calculation?
A: The calculation is mathematically precise when using the exact formula. The calculator provides results with 6 decimal places for practical accuracy.

Q3: Can I use different units of measurement?
A: The formula works with any consistent unit of length, but the calculator is set up for meters. You can convert your measurements to meters before calculation.

Q4: What are the applications of this calculation?
A: This calculation is useful in geometry education, architectural design, 3D modeling, and any field dealing with polyhedral structures.

Q5: Is the truncated dodecahedron a common shape in nature?
A: While not as common as simpler polyhedra, truncated dodecahedra appear in some crystal structures and molecular models, particularly in fullerene molecules.