Edge Length of Truncated Dodecahedron given Midsphere Radius Calculator

Edge Length of Truncated Dodecahedron given Midsphere Radius Formula:

\[ l_e = \frac{4 \times r_m}{5 + (3 \times \sqrt{5})} \]

Edge Length of Truncated Dodecahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Edge Length of Truncated Dodecahedron given Midsphere Radius?

The edge length of a truncated dodecahedron given its midsphere radius is a geometric calculation that determines the length of any edge of the polyhedron based on the radius of the sphere that is tangent to all its edges.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_e = \frac{4 \times r_m}{5 + (3 \times \sqrt{5})} \]

Where:

\( l_e \) — Edge Length of Truncated Dodecahedron (m)
\( r_m \) — Midsphere Radius of Truncated Dodecahedron (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula establishes the mathematical relationship between the midsphere radius and the edge length of a truncated dodecahedron, taking into account the geometric properties of this Archimedean solid.

3. Importance of Edge Length Calculation

Details: Calculating the edge length from the midsphere radius is crucial for geometric modeling, architectural design, and understanding the spatial properties of truncated dodecahedrons in various applications.

4. Using the Calculator

Tips: Enter the midsphere radius in meters. The value must be positive and valid. The calculator will compute the corresponding edge length of the truncated dodecahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a truncated dodecahedron?
A: A truncated dodecahedron is an Archimedean solid created by truncating the vertices of a regular dodecahedron, resulting in 20 regular triangular faces and 12 regular decagonal faces.

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

Q3: What are typical values for edge lengths?
A: Edge lengths vary depending on the size of the polyhedron, but they are typically in the range of centimeters to meters for practical applications.

Q4: Can this formula be used for other polyhedra?
A: No, this specific formula applies only to truncated dodecahedrons. Other polyhedra have different geometric relationships.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of the truncated dodecahedron, provided the input values are accurate.