Insphere Radius of Icosahedron given Face Area Calculator

Formula Used:

\[ r_i = \frac{\sqrt{3} \cdot (3 + \sqrt{5})}{12} \cdot \sqrt{\frac{4 \cdot A_{Face}}{\sqrt{3}}} \]

Insphere Radius of Icosahedron:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Insphere Radius of Icosahedron?

The Insphere Radius of Icosahedron is the radius of the sphere that is contained by the Icosahedron in such a way that all the faces just touching the sphere. It represents the largest sphere that can fit inside the icosahedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\sqrt{3} \cdot (3 + \sqrt{5})}{12} \cdot \sqrt{\frac{4 \cdot A_{Face}}{\sqrt{3}}} \]

Where:

\( r_i \) — Insphere Radius of Icosahedron (m)
\( A_{Face} \) — Face Area of Icosahedron (m²)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)
\( \sqrt{5} \) — Square root of 5 (approximately 2.236)

Explanation: This formula calculates the radius of the inscribed sphere based on the area of one face of the icosahedron, utilizing geometric properties of regular polyhedra.

3. Importance of Insphere Radius Calculation

Details: Calculating the insphere radius is important in geometry, material science, and engineering applications where understanding the internal dimensions and packing efficiency of polyhedral structures is required.

4. Using the Calculator

Tips: Enter the face area of the icosahedron in square meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is an icosahedron?
A: An icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 12 vertices, and 30 edges.

Q2: How is face area related to insphere radius?
A: The face area determines the size of the icosahedron, which in turn determines the size of the largest sphere that can fit inside it (insphere).

Q3: Can this formula be used for irregular icosahedrons?
A: No, this formula is specifically for regular icosahedrons where all faces are identical equilateral triangles.

Q4: What are practical applications of this calculation?
A: This calculation is used in crystallography, molecular modeling, architectural design, and various engineering fields involving polyhedral structures.

Q5: How accurate is this formula?
A: The formula is mathematically exact for perfect regular icosahedrons and provides precise results when accurate input values are provided.