Long Ridge Length of Great Icosahedron Given Short Ridge Length Calculator

Formula Used:

\[ l_{Ridge(Long)} = \frac{\sqrt{2} \cdot (5 + 3\sqrt{5})}{10} \cdot \frac{5 \cdot l_{Ridge(Short)}}{\sqrt{10}} \]

Long Ridge Length of Great Icosahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Long Ridge Length of Great Icosahedron?

The Long Ridge Length of Great Icosahedron is the length of any of the edges that connects the peak vertex and adjacent vertex of the pentagon on which each peak of Great Icosahedron is attached. It is an important geometric measurement in the study of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ l_{Ridge(Long)} = \frac{\sqrt{2} \cdot (5 + 3\sqrt{5})}{10} \cdot \frac{5 \cdot l_{Ridge(Short)}}{\sqrt{10}} \]

Where:

\( l_{Ridge(Long)} \) — Long Ridge Length of Great Icosahedron (meters)
\( l_{Ridge(Short)} \) — Short Ridge Length of Great Icosahedron (meters)
\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)
\( \sqrt{5} \) — Square root of 5 (approximately 2.2361)
\( \sqrt{10} \) — Square root of 10 (approximately 3.1623)

Explanation: This formula establishes the precise mathematical relationship between the long ridge length and short ridge length of a Great Icosahedron, incorporating fundamental mathematical constants and geometric relationships.

3. Importance of Long Ridge Length Calculation

Details: Accurate calculation of the long ridge length is crucial for geometric modeling, architectural design, and mathematical analysis of the Great Icosahedron. It helps in understanding the spatial relationships and proportions within this complex polyhedral structure.

4. Using the Calculator

Tips: Enter the short ridge length in meters. The value must be a positive number greater than zero. The calculator will automatically compute the corresponding long ridge length using the mathematical formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a Great Icosahedron?
A: The Great Icosahedron is one of the four Kepler-Poinsot polyhedra, a non-convex regular polyhedron with 20 triangular faces that intersect each other.

Q2: What are the practical applications of this calculation?
A: This calculation is primarily used in mathematical geometry, architectural design, and 3D modeling where precise measurements of complex polyhedra are required.

Q3: How accurate is this formula?
A: The formula provides mathematically exact results based on the geometric properties of the Great Icosahedron, making it highly accurate for theoretical calculations.

Q4: Can this calculator be used for other polyhedra?
A: No, this specific formula and calculator are designed specifically for the Great Icosahedron and its ridge length relationships.

Q5: What units should I use for input?
A: The calculator accepts input in meters, but you can use any consistent unit of length as the relationship is proportional.